arXiv:quant-ph/0702225v2 20 Apr 2007 unu entanglement Quantum Contents I.Poern ffcsbsdo entanglement on based effects Pioneering III. I iatt entanglement Bipartite VI. V orlto aiettoso entanglement: of manifestations Correlation IV. I nageeta unu rpryof property quantum a as Entanglement II. .Etoi aiettoso entanglement of manifestations Entropic V. .Introduction I. elinequalities. Bell opudsystems compound .Bl hoe eodCS-etn 15 CHSH-setting beyond theorem Bell D. .Etnlmn wpig12 swapping Entanglement D. .Dfiiinadbscpoete 20 properties basic and Definition A. .Etoi nqaiis lsia essquantum versus classical inequalities: Entropic A. .Bl hoe:CS nqaiy 13 inequality. CHSH theorem: Bell A. .Qatmkydsrbto ae netnlmn 10 entanglement on based distribution key Quantum A. .Mjrzto eain 20 relations Majorization C. .Nnoaiyo unu ttsadLVmdl14 model LHV and states quantum of Nonlocality C. .Qatmtlprain11 teleportation Quantum C. .Mi eaaiiyetnlmn rtrain criteria separability/entanglement Main B. .1etoi nqaiisadngtvt of negativity and inequalities 1-entropic B. .TeotmlCS nqaiyfr2 for inequality CHSH optimal The B. .Qatmdnecdn 10 coding dense Quantum B. .Lgclvrin fBl’ hoe 16 theorem Bell’s of versions Logical E. .Baigcasclcmuiaincomplexity communication classical Beating E. .Voaino elieulte:gnrlrmrs17 remarks general inequalities: Bell of Violation F. yzr Horodecki Ryszard 1 2 l u omreprec ihapiaino unu theor quantum of application with experience former our All 3 ai oeo nageetwtessi eeto fentang of detection in witnesses b pec entanglement — of some form rol role stress extremal basic basic its and a including paradigm discuss manipulations labs entanglement also distant They within interrelations. munication entangleme some inequalities, out discuss entropic point authors inequalities, the Bell particular, via In includi entanglement quantifying. of whic and aspects of lation basic task usuall reviews the Being article tools, This detect. mathematical to and difficult conceptual and against complex coding. very dense processes is and quantum resource many teleportation quantum for cryptography, potential i tum a which energy. is systems, as quantum subsystems, real compound tween as of resource property holistic new Schr¨odiThis a and as Rosen laboratories Podolsky, to Einstein, by laboratory recognized in ment, occur must formalism quantum nttt fTertclPyisadAtohsc Univers Astrophysics and Physics Theoretical of Institute aut fMteais hsc n optrSineUnivers Science Computer and Physics Mathematics, of Faculty aut fApidPyisadMteais ehia Unive Technical Mathematics, and Physics Applied of Faculty iatt case bipartite information order onswt nageet12 entanglement with bounds .Sprblt i nageetwtess22 of detection experimental and Witnesses witnesses entanglement 4. via 21 Separability 3. criterion Separability (PPT) transpose 2. partial Positive 1. .Mxdsae 14 states Mixed 2. .Pr states Pure 1. nageet24 21 entanglement maps positive via 1 aelHorodecki Pawe l oiie u o completely not but positive, × ytm 14 systems 2 3 u h sec fqatmfraim—entangle- — formalism quantum of essence the But . ih Horodecki Micha l twtess unu rporpyand cryptography quantum witnesses, nt 21 20 19 18 18 14 13 10 8 2 giscaatrzto,dtcin distil- detection, characterization, its ng aiu aiettoso entanglement of manifestations various eeti emphasized. is lement gr—wie vr7 er oenter to years 70 over waited — nger st eihrisrc structure. rich its decipher to is h fetnlmn nqatmcom- quantum in entanglement of e rgl oevrnet ti robust is it environment, to fragile y em osay: to seems y t fGank 092Gda´nsk, Poland Gda´nsk, 80–952 of ity udetnlmn hnmnn A phenomenon. entanglement ound vle ocasclcreain be- correlations nonclassical nvolves oee,i perdta hsnew this that appeared quan- it However, ones: “canonical” including , laiissc sirvriiiyof irreversibility as such uliarities II ute mrvmnso nageettests: entanglement of improvements Further VIII. I.Mliatt nageet—smlrte and similarities — entanglement Multipartite VII. I.Dsilto n on entanglement bound and Distillation XII. I h aaimo oa prtosadclassical and operations local of paradigm The XI. entanglement detecting algorithms Classical IX. 1 st fGank 092Gda´nsk, Poland Gda´nsk, 80–952 of rsity .Qatmetnlmn n geometry and entanglement Quantum X. ao Horodecki Karol , t fGank 092Gank oadand Gda´nsk, Gda´nsk, Poland 80–952 of ity differences olna eaaiiycriteria separability nonlinear omncto (LOCC) communication .Nto ffl ( full of Notion A. .Oewyhsigdsilto rtcl41 protocol distillation hashing One-way A. .Ucranyrlto ae eaaiiytss33 tests separability based relation Uncertainty A. .Qatmcanl—temi oin39 notion main the — channel Quantum A. .Dtcigetnlmn ihcollective with entanglement Detecting C. .Prilsprblt 32 separability Partial B. .Towyrcrec itlainpooo 42 protocol distillation recurrence Two-way B. .Nnieripoeeto nageetwtess34 witnesses entanglement of improvement Nonlinear B. .LC prtos39 operations LOCC B. 1 nacn eaaiiyciei ylclfitr 30 filters local by criteria separability Enhancing 11. in separability bipartite of Characterization 10. measurements .Sm lse fipratqatmstates: quantum important of classes Some 9. linear and criterion realignment Matrix PPT 8. applications; its and criterion Range 7. 26 reduction criteria: maps inequalities Distinguished Bell and 6. witnesses Entanglement 5. .Cletv nageetwtess36 as entanglement quantum of Detection witnesses 3. entanglement Collective 2. entanglement of implementations Physical 1. hti rdce by predicted is what em fbcnurne29 26 28 29 26 biconcurrence of terms parameters of regions entanglement criteria contractions entanglement extensions its and criterion rtrawt olciemaueet 35 data structure quantum with computing quantum measurements collective with criteria 1 , m 2 prie eaaiiy30 separability -partite) 30 41 33 37 39 37 38 35 2
C. Development of distillation protocols — bipartite A. Pure states 73 and multipartite case 42 B. Mixed states 74 D. All two-qubit entangled states are distillable 43 C. Gaussian entanglement 74 E. Reduction criterion and distillability 44 D. General separability criteria for continuous F. General one-way hashing 44 variables 76 G. Bound entanglement — when distillability fails 44 E. Distillability and entanglement measures of H. The problem of NPT bound entanglement 45 Gaussian states 77 I. Activation of bound entanglement 45 1. Multipartite bound entanglement 47 XVIII. Miscellanea 77 J. Bell inequalities and bound entanglement 47 A. Entanglement under information loss: locking entanglement 77 XIII. Manipulations of entanglement and B. Entanglement and distinguishing states by LOCC 79 irreversibility 48 C. Entanglement and thermodynamical work 80 A. LOCC manipulations on pure entangled states — D. Asymmetry of entanglement 81 exact case 48 1. Entanglement catalysis 48 XIX. Entanglement and secure correlations 81 2. SLOCC classification 48 A. Quantum key distribution schemes and security B. Asymptotic entanglement manipulations and proofs based on distillation of pure entanglement 81 irreversibility 49 1. Entanglement distillation based quantum key 1. Unit of bipartite entanglement 49 distribution protocols. 82 2. Bound entanglement and irreversibility 50 2. Entanglement based security proofs 83 3. Asymptotic transition rates in multipartite 3. Constraints for security from entanglement 84 states 50 4. Secure key beyond distillability - prelude 84 B. Drawing private key from distillable and bound XIV. Entanglement and quantum communication 51 n A. Capacity of quantum channel and entanglement 52 entangled states of the form ρ⊗ 84 B. Fidelity of teleportation via mixed states 52 1. Drawing key from distillable states: C. Entanglement breaking and entanglement binding Devetak-Winter protocol 85 channels 53 2. Private states 85 D. Quantum Shannon theorem 53 3. Private states versus singlets 86 E. Bell diagonal states and related channels 54 4. Purity and correlations: how they are present F. Other capacities of quantum channels 54 in p-bit 86 G. Additivity questions 55 5. Distillable key as an operational entanglement H. Miscellanea 55 measure 87 6. Drawing secure key from bound entanglement. 87 XV. Quantifying entanglement 56 C. Private states — new insight into entanglement A. Distillable entanglement and entanglement cost 56 theory of mixed states 88 B. Entanglement measures — axiomatic approach 57 D. Quantum key distribution schemes and security 1. Monotonicity axiom 57 proofs based on distillation of private states - 2. Vanishing on separable states. 58 private key beyond purity 88 3. Other possible postulates. 58 1. “Twisting” the standard protocol 88 4. Monotonicity for pure states. 58 E. Entanglement in other cryptographic scenarios 89 5. Monotonicity for convex functions 58 1. Impossibility of quantum bit commitment — 6. Invariance under local unitary transformations 59 when entanglement says no 89 C. Axiomatic measures — a survey 59 2. Multipartite entanglement in quantum secret 1. Entanglement measures based on distance 59 sharing 89 2. Convex roof measures 60 3. Other multipartite scenarios 90 3. Mixed convex roof measures 62 F. Interrelations between entanglement and classical 4. Other entanglement measures 62 key agreement 90 D. All measures for pure bipartite states 64 1. Classical key agreement — analogy to 1. Entanglement measures and transition between distillable entanglement scenario 91 states — exact case 65 2. Is there a bound information? 92 E. Entanglement measures and transition between states — asymptotic case 65 XX. Entanglement and quantum computing 92 1. ED and EC as extremal measures. Unique A. Entanglement in quantum algorithms 92 measure for pure bipartite states. 65 B. Entanglement in quantum architecture 93 2. Transition rates 66 C. Byzantine agreement — useful entanglement for F. Evaluating measures 66 quantum and classical distributed computation 94 1. Hashing inequality 67 2. Evaluating EC vs additivity problem 67 ACKNOWLEDGMENTS 94 G. Entanglement imposes different orderings 68 H. Multipartite entanglement measures 68 References 95 1. Multipartite entanglement measures for pure states 69 I. Entanglement parameters 71 J. How much can entanglement increase under I. INTRODUCTION communication of one qubit? 71 Although in 1932 von Neumann had completed ba- XVI. Monogamy of entanglement 72 sic elements of nonrelativistic quantum description of XVII. Entanglement in continuous variables systems 73 the world, it were Einstein, Podolsky and Rosen (EPR) 3 and Schr¨odinger who first recognized a “spooky” fea- these experiments strongly confirmed the predictions of ture of quantum machinery which lies at center of in- the quantum description1. terest of physics of XXI century (Einstein et al., 1935; In fact, a fundamental nonclassical aspect of entan- von Neumann, 1932). This feature implies the existence glement was recognized already in 1935. Inspired by of global states of composite system which cannot be writ- EPR paper, Shr¨odinger analyzed some physical conse- ten as a product of the states of individual subsystems. quences of quantum formalism and he noticed that the This phenomenon, known as “entanglement”, was origi- two-particle EPR state does not admit ascribing indi- nally called by Schr¨odinger “Verschr¨ankung”, which un- vidual states to the subsystems implying “entanglement derlines an intrinsic order of statistical relations between of predictions” for the subsystems. Then he concluded: subsystems of compound quantum system (Schr¨odinger, “Thus one disposes provisionally (until the entanglement 1935). is resolved by actual observation) of only a common de- Paradoxically, entanglement, which is considered to be scription of the two in that space of higher dimension. the most nonclassical manifestations of quantum formal- This is the reason that knowledge of the individual sys- ism, was used by Einstein Podolsky and Rosen in their tems can decline to the scantiest, even to zero, while attempt to ascribe values to physical quantities prior to that of the combined system remains continually maxi- measurement. It was Bell who showed the opposite: it mal. Best possible knowledge of a whole does not include is just entanglement which irrevocably rules out such a best possible knowledge of its parts — and this is what possibility. keeps coming back to haunt us”(Schr¨odinger, 1935)2. In 1964 Bell accepted the EPR conclusion — that Unfortunately this curious aspect of entanglement quantum description of physical reality is not complete was long unintelligible, as it was related to the no- — as a working hypothesis and formalized the EPR de- tion of “knowledge” in quantum context. Only in half terministic world idea in terms of local hidden variable of 90’s it was formalized in terms of entropic inequali- model (LHVM) (Bell, 1964). The latter assumes that ties based on Von Neumann entropy (Cerf and Adami, (i) measurement results are determined by properties the 1997; Horodecki and Horodecki, 1994; Horodecki et al., particles carry prior to, and independent of, the measure- 1996c)3. The violation of these inequalities by entangled ment (“realism”), (ii) results obtained at one location are states is a signature of entanglement of quantum states, independent of any actions performed at spacelike sepa- however physical meaning of this was unclear. An in- ration (“locality”) (iii) the setting of local apparatus are teresting attempt to solve this puzzle is due to Cerf and independent of the hidden variables which determine the Adami (1997) in terms of conditional entropy. Soon af- local results (“free will”). Bell proved that the above as- terwards it turned out that the latter with minus sign, sumptions impose constraints on statistical correlations called coherent information is a fundamental quantity re- in experiments involving bipartite systems in the form of sponsible for capabilities of transmission of quantum in- the Bell inequalities. He then showed that the probabili- formation (Lloyd, 1997; Schumacher and Nielsen, 1996). ties for the outcomes obtained when suitably measuring The transmission is possible exactly in those situations some entangled quantum state violate the Bell inequality. in which “Schr¨odinger’s demon” is “coming to haunt us” In this way entanglement is that feature of quantum for- — i.e. when entropy of output system exceeds the en- malism which makes impossible to simulate the quantum tropy of the total system. Let us mention, that in 2005 correlations within any classical formalism. this story has given a new twist in terms of quantum Greenberger, Horne and Zeilinger (GHZ) went be- counterpart of the Slepian-Wolf theorem in classical com- yond Bell inequalities by showing that entanglement of munication (Horodecki et al., 2005h, 2006e). In this ap- more than two particles leads to a contradiction with proach the violation of entropic inequalities implies the LHVM for nonstatistical predictions of quantum formal- existence of negative quantum information, which is “ex- ism (Greenberger et al., 1989). Surprisingly, only in the tra” resource for quantum communication. Interestingly, beginning of 90’s theoretical general results concerning only recently a direct violation of the entropic inequalities violation of Bell inequalities have been obtained (Gisin, was experimentally demonstrated confirming the break- 1991; Popescu and Rohrlich, 1992). ing of classical statistical order in compound quantum Transition of entanglement from gedanken exper- systems (Bovino et al., 2005). iment to laboratory reality began in the mid-60s The present-day entanglement theory has its roots (Freedman and Clauser, 1972; Kocher and Commins, in the key discoveries: quantum cryptography with 1967). However it were Aspect et al., who performed first a convincing test of violation of the Bell inequalities (Aspect et al., 1982, 1981). Since then many kinds of beautiful and precise experimental tests of quantum 1 However so far all the above experiments suffer from loophole, formalism against the LHVM have been performed in see (Brunner et al., 2007; Gill, 2003). 2 laboratories (Bovino et al., 2006a; Hasegawa et al., an English translation appears in Quantum Theory and Mea- surement, edited by J. A. Wheeler and W.H. Zurek Princeton 2003; Kwiat et al., 1995; Ou and Mandel, 1988; University Press, Princeton, 1983, p.167. Rowe and et al., 2001) and outsides (Tittel et al., 3 The other formalization was proposed in terms of majorization 1998, 1999; Ursin et al., 2006; Weihs et al., 1998). All relations (Nielsen and Kempe, 2001). 4
Bell theorem (Ekert, 1991), quantum dense coding for understanding many physical phenomena including (Bennett and Wiesner, 1992) and quantum teleporta- super-radiance (Lambert et al., 2004), superconductiv- tion (Bennett et al., 1993)4 including teleportation of ity (Vedral, 2004), disordered systems (D¨ur et al., 2005) entanglement of EPR pairs (so called entanglement and emerging of classicality (Zurek, 2003a). In partic- swapping) (Bose et al., 1998; Yurke and Stoler, 1992a,b; ular, understanding the role of entanglement in the ex- Zukowski˙ et al., 1993). All such effects are based isting methods of simulations of quantum spin systems on entanglement and all of them have been demon- allowed for significant improvement of the methods, as strated in pioneering experiments (see (Boschi et al., well as understanding their limitations (Anders et al., 1998; Bouwmeester et al., 1997; Furusawa et al., 1998; 2006; Verstraete et al., 2004b; Vidal, 2003, 2004). Jennewein et al., 2000; Mattle et al., 1996b; Naik et al., The role of entanglement in quantum phase tran- 2000; Pan et al., 1998; Tittel et al., 2000). In fact, sitions (Larsson and Johannesson, 2006; Latorre et al., the above results including pioneering paper on 2004; Osborne and Nielsen, 2002; Osterloh et al., 2002; quantum cryptography (Bennett and Brassard, 1984) Verstraete et al., 2004a; Vidal et al., 2003) was inten- and idea of quantum computation (Deutsch, 1985; sively studied. Divergence of correlations at critical Feynman, 1982; Shor, 1995; Steane, 1996a) were a ba- points is always accompanied by divergence of a suitably sis for a new interdisciplinary domain called quantum defined entanglement length (Verstraete et al., 2004a). information (Alber et al., 2001a; Bouwmeester et al., The concept of entanglement length originates form ear- 2000; Braunstein and Pati, 2003; Bruß and Leuchs, 2007; lier paper by D. Aharonov, where a critical phenomenon Lo et al., 1999; Nielsen and Chuang, 2000) which incor- has been studied in context of fault-tolerant quantum porated entanglement as a central notion. computing (Aharonov, 1999). It has become clear that entanglement is not only sub- Unfortunately quantum entanglement has three dis- ject of philosophical debates, but it is a new quantum agreeable but interesting features: It has in general very resource for tasks which can not be performed by means complex structure, it is very fragile to environment and it of classical resources (Bennett, 1998). It can be manip- can not be increased on average when systems are not in ulated (Bennett et al., 1996c,d; Gisin, 1996b; Popescu, direct contact but distributed in spatially separated re- 1995; Raimond et al., 2001), broadcasted (Buzek et al., gions. The theory of entanglement tries to give answers 1997), controlled and distributed (Beige et al., 2000; to fundamental questions as: i) How to detect optimally Cirac and Zoller, 2004; Mandilara et al., 2007). entanglement theoretically and in laboratory; ii) How to Remarkably, entanglement is a resource which, though reverse an inevitable process of degradation of entangle- does not carry information itself, can help in such ment; iii) How to characterize, control and quantify en- tasks as the reduction of classical communication com- tanglement. plexity (Brukner et al., 2004; Buhrman et al., 2001a; History of the questions i) and ii) has its origin in Cleve and Buhrman, 1997), entanglement assisted ori- seminal papers by Werner and Popescu (Popescu, 1995; entation in space (Bovino et al., 2006b; Brukner et al., Werner, 1989a). Werner not only gave accurate definition 2005), quantum estimation of a damping constant of separable states (those mixed states that are not en- (Venzl and Freyberger, 2007), frequency standards im- tangled), but also noted that there exist entangled states provement (Giovannetti et al., 2004; Huelga et al., 1997; that similarly to separable states, admit LHV model, Wineland et al., 1992) (see in this context (Boto et al., hence do not violate Bell inequalities. Popescu showed 2005)) and clock synchronization (Jozsa et al., 2000). (Popescu, 1995) that having system in such state, by The entanglement plays fundamental role in quantum means of local operations one can get a new state whose communication between parties separated by macro- entanglement can be detected by Bell inequalities. This scopic distances (Bennett et al., 1996d). idea was developed by Gisin who used so-called filters to Though the role of entanglement in quantum computa- enhance violating of Bell inequalities (Gisin, 1996b). In tional speed-up is not clear (Kendon and Munro, 2006a), fact, this idea turned out to be a trigger for a theory of it has played important role in development of quan- entanglement manipulations (Bennett et al., 1996c). tum computing, including measurement based schemes, one-way quantum computing (Raussendorf and Briegel, Soon afterwards, Peres showed that if the state was 7 8 2001)5, linear optics quantum computing (Knill et al., separable then after partial transpose of density ma- 2001)6. Entanglement has also given us new insights trix on one of the subsystems of a compound bipartite system it is still a legitimate state (Peres, 1996a). Sur- prisingly Peres condition appeared to be strong test for
4 Quantum teleportation with continuous variables in an infi- nite dimensional Hilbert space was first proposed by Vaidman (Vaidman, 1994) and further investigated theoretically by Braun- 2006). stein and Kimble (Braunstein and Kimble, 1998). 7 More formal definitions of entangled and separable states are 5 For comprehensive review see (Browne and Briegel, 2006). given in the next section. 6 It has been shown that linear optics quantum computing can 8 The positive partial transpose condition is called the Peres cri- viewed as a measurement based one-way computing (Popescu, terion or the PPT criterion of separability. 5
entanglement9. As the partial transpose is a positive 2003; Bourennane et al., 2004; Haffner et al., 2005; map it was realized that the positive maps can serve Leibfried et al., 2005; Lu et al., 2007; Mikami et al., as the strong detectors of entanglement. However they 2005; Resch et al., 2005; Roos et al., 2004). cannot be implemented directly in laboratory, because As one knows, the main virtue of entanglement wit- they are unphysical10. Fortunately there is “footbridge”- nesses is that they provide an economic way of detection Jamio lkowski isomorphism (Jamio lkowski, 1972a) which of entanglement, that does not need full (tomographic) allowed to go to physical measurable quantities — Her- information about the state. It arises a natural ques- mitian’s operators. This constitutes a necessary and tion: how to estimate optimally the amount of entan- sufficient condition separability on the both, physical glement of compound system in an unknown state if level of observables and nonphysical one engaging pos- only incomplete data in the form of averages values of itive maps (Horodecki et al., 1996a). This characteriza- some operators detecting entanglement are accessible? tion of entanglement although nonoperational, provides This question posed in 1998 lead to the new inference a basis for general theory of detection of entanglement. scheme for all the processes where entanglement is mea- A historical note is here in order. Namely it turned out sured. It involves a principle of minimization of entan- that both the general link between separability and pos- glement under a chosen measure of entanglement with itive maps as well as the Peres-Horodecki theorem were constrains in the form of incomplete set of data from ex- first known and expressed in slightly different language periment (Horodecki et al., 1999c). In particular, mini- as early as in 70s (Choi, 1972) (see also (Osaka, 1991; mization of entanglement measures (entanglement of for- Størmer, 1963; Woronowicz, 1976)). The rediscovery by mation and relative entropy of entanglement) under fixed Peres and Horodecki’s brought therefore powerful meth- Bell-type witness constraint was obtained. Subsequently ods to entanglement theory as well as caused revival of the inference scheme based of the minimization of entan- research on positive maps, especially the so-called non- glement was successfully applied (Audenaert and Plenio, decomposable ones. 2006; Eisert et al., 2007; G¨uhne et al., 2006b) to estimate Terhal was the first to construct a family of inde- entanglement measures from the results of recent exper- composable positive linear maps based on entangled iments measuring entanglement witnesses. This results quantum states (Terhal, 2000b). She also pointed show clearly that the entanglement witnesses are not only out, that a violation of a Bell inequality can for- economic indicators of entanglement but they are also mally be expressed as a witness for entanglement helpful in estimating the entanglement content. (Terhal, 2000a)11. Since then theory of entangle- In parallel to entanglement witnesses the theory ment witnesses was intensively developed (Brandao, of positive maps was developed which provides, in 2005; Brandao and Vianna, 2006; Bruß et al., 2002; particular strong tools for the detection of entangle- G¨uhne et al., 2003; Kiesel et al., 2005; Lewenstein et al., ment (Benatti et al., 2004; Breuer, 2006a; Cerf et al., 2000; Toth and G¨uhne, 2005a), including a non- 1999; Chru´sci´nski and Kossakowski, 2006; Datta et al., linear generalization (G¨uhne and Lewenstein, 2004b; 2006b; Hall, 2006; Horodecki and Horodecki, 1999; G¨uhne and L¨utkenhaus, 2006a) and the study of indistin- Kossakowski, 2003; Majewski, 2004; Piani, 2004; guishable systems (Eckert et al., 2002; Schliemann et al., Piani and Mora, 2006; Terhal, 2000b). Strong insepara- 2001). bility criteria beyond the positive maps approach were The concept of entanglement witness was applied to also found (Chen and Wu, 2003; Clarisse and Wocjan, different problems in statistical systems (Brukner et al., 2006; Devi et al., 2007; G¨uhne, 2004; G¨uhne et al., 2007; 2006; Cavalcanti et al., 2006; Wie´sniak et al., 2005; Hofmann and Takeuchi, 2003; Horodecki et al., 2006d; Wu et al., 2005), quantum cryptography (Curty et al., Mintert et al., 2005b; Rudolph, 2000). 2005) quantum optics (Stobi´nska and W´odkiewicz, 2005; Separability criteria for continuous variables were also Stobi´nska and W´odkiewicz, 2006), condensed-matter proposed (see Sec. XVII.D). The necessary and sufficient nanophysics (Blaauboer and DiVincenzo, 2005), bound condition for separability of Gaussian states of a bipartite entanglement (Hyllus et al., 2004), experimental real- system of two harmonic oscillators has been found inde- ization of cluster states (Vallone et al., 2007), hid- pendently by Simon and Duan et al. (Duan et al., 2000; den nonlocality (Masanes et al., 2007). About this Simon, 2000). Both approaches are equivalent. Simon’s time in precision experiments a detection of multi- criterion is direct generalization of partial transpose to partite entanglement using entanglement witness oper- CV systems, while Duan et al. started with local uncer- ators was performed (Altepeter, 2005; Barbieri et al., tainty principles. Soon afterwards (Werner and Wolf, 2001a) found bound entangled Gaussian states. Since then theory of entanglement continuous variables have been developed 9 For low dimensional systems it turned out to be necessary and in many directions, especially for Gaussian states which sufficient condition of separability what is called Peres-Horodecki are accessible at the current stage of technology (see criterion (Horodecki et al., 1996a). 10 See however (Horodecki and Ekert, 2002). (Braunstein and Pati, 2003) and references therein). For 11 The term “entanglement witness” for operators detecting entan- the latter states the problem of entanglement versus sep- glement was introduced in (Terhal, 2000a). arability was solved completely: operational necessary 6 and sufficient condition was provided in (Giedke et al., there exists bound entanglement. It destroyed the hope 2001b). This criterion therefore detects all bipartite that noisy entanglement can have more or less uniform bound entangled Gaussian states. Interestingly McHugh structure: instead we encounter peculiarity in the struc- et al. constructed a large class of non-Gaussian two-mode ture of entanglement. Namely there is free entanglement continuous variable states for which separability crite- that can be distilled and the bound one - very weak form rion for Gaussian states can be employed (McHugh et al., of entanglement. Passivity of the latter provoked inten- 2007). sive research towards identifying any tasks that would re- Various criteria for continuous variables were ob- veal its quantum features. More precisely any entangled tained (Agarwal and Biswas, 2005; Hillery and Zubairy, state that has so called positive partial transpose cannot 2006; Mancini et al., 2002; Raymer et al., 2003). A be distilled. First explicit examples of such states were powerful separability criterion of bipartite harmonic provided in (Horodecki, 1997). Further examples (called quantum states based on partial transpose was de- bound entangled) were found in (Bennett et al., 1999b; rived which includes all the above criteria as a special Bruß and Peres, 2000; Werner and Wolf, 2001a); see also cases (Miranowicz and Piani, 2006; Shchukin and Vogel, (Bruß and Leuchs, 2007; Clarisse, 2006b) and references 2005b). therein. The present day entanglement theory owes its form in Existence of bound entanglement has provoked many great measure to the discovery of the concept of entan- questions including relations to local variable model and glement manipulation (Bennett et al., 1996c; Popescu, role as well as long standing and still open question of 1995). It was realized (Bennett et al., 1996c) that a existence of bound entangled states violating PPT crite- natural class of operations suitable for manipulating rion, which would have severe consequences for commu- entanglement is that of local operations and classical nication theory. communication (LOCC), both of which cannot bring in Due to bound entanglement, the question can every en- entanglement for free. So established distant lab (or tanglement be used for some useful quantum task? had LOCC) paradigm plays a fundamental role in entangle- stayed open for a long time. Only quite recently positive ment theory. Within the paradigm many important re- answer both for bipartite as well as multipartite states sults have been obtained. In particular, the framework was given by Masanes (Masanes, 2005a,b) in terms of for pure state manipulations have been established, in- so-called activation (Horodecki et al., 1999a). This re- cluding reversibility in asymptotic transitions of pure markable result allows to define entanglement not only bipartite states (Bennett et al., 1996b), connection be- in negative terms of Werner’s definition (a state is en- tween LOCC pure state transitions and majorization the- tangled if it is not mixture of product states) but also in ory (Schumacher and Nielsen, 1996) as well as a sur- positive terms: a state is entangled, if it is a resource for prising effect of catalysis (Jonathan and Plenio, 1999; a nonclassical task. Vidal and Cirac, 2001, 2002). Moreover, inequivalent One of the most difficult and at the same time fun- types of multipartite entanglement have been identified damental questions in entanglement theory, is quan- (Bennett et al., 2001; D¨ur et al., 2000b). tifying entanglement. Remarkably, two fundamen- Since in laboratory one usually meets mixed states rep- tal measures, entanglement distillation (Bennett et al., resenting noisy entanglement, not much useful for quan- 1996b,c,d), and what is now called entanglement cost tum information processing, there was a big challenge: (Bennett et al., 1996d; Hayden et al., 2001), appeared to reverse the process of degradation of entanglement in the context of manipulating entanglement, and have by means of some active manipulations. Remarkably in an operational meaning. Their definitions brings in (Bennett et al., 1996c) it was shown that it is possible to mind thermodynamical analogy (Horodecki et al., 1998b; distill pure entanglement from noisy one in asymptotic Popescu and Rohrlich, 1997; Vedral and Plenio, 1998), regime. It should be noted that in parallel, there was since they describe two opposite processes — creation intense research aiming at protection of quantum infor- and distillation, which ideally should be reverse of each mation in quantum computers against decoherence. As a other. Indeed, reversibility holds for pure bipartite result error correcting codes have been discovered (Shor, states, but fails for noisy as well as multipartite entangle- 1995; Steane, 1996a). Very soon it was realized that ment. The generic gap between these two measures shows quantum error correction and distillation of entangle- a fundamental irreversibility (Horodecki et al., 1998a; ment are in fact inherently interrelated (Bennett et al., Vidal and Cirac, 2001; Yang et al., 2005a). This phe- 1996d). nomenon has its origin deeply in the nature of noisy en- A question fundamental for quantum information pro- tanglement, and its immediate consequence is the non- cessing has then immediately arisen: “Can noisy en- existence of a unique measure of entanglement. tanglement be always purified?”. A promising result Vedral et al. (Vedral and Plenio, 1998; Vedral et al., was obtained in (Horodecki et al., 1997), where all two- 1997b) have proposed an axiomatic approach to quan- qubit noisy entanglement were showed to be distill- tifying entanglement, in which a “good” measure of en- able. However, soon afterwards a no-go type result tanglement is any function that satisfy some postulates. (Horodecki et al., 1998a) has revealed dramatic differ- The leading idea (Bennett et al., 1996d) is that entan- ence between pure and noisy entanglement: namely, glement should not increase under local operations and 7
classical communication so called monotonicity condi- 2004; Ficek and Tana´s, 2006; Jak´obczyk and Jamr´oz, tion. The authors proposed a scheme of obtaining based 2004; Kim et al., 2002; Maloyer and Kendon, 2007; on the concept on distance from separable states, and Mintert et al., 2005a; Miranowicz, 2004a; Montangero, introduced one of the most important measures of en- 2004; Shresta et al., 2005; Wang et al., 2006; Yi and Sun, tanglement, the so called relative entropy of entangle- 1999; Yu and Eberly, 2004; Zyczkowski˙ et al., 2002) en- ment (Vedral and Plenio, 1998; Vedral et al., 1997b) (for tanglement production, in the course of quantum compu- more comprehensive review see (Vedral, 2002)). Subse- tation (Kendon and Munro, 2006b; Parker and Plenio, quently, a general mathematical framework for entangle- 2002) or due to interaction between subsystems. The ment measures was worked out by Vidal (Vidal, 2000), latter problem gave rise to a notion of “entan- who concentrated on the axiom of monotonicity (hence gling power” (Linden et al., 2005; Zanardi et al., 2000) the often used term “entanglement monotone”). of a unitary transformation, which can be seen as At first sight it could seem that measures of entan- a higher-level entanglement theory dealing with en- glement do not exhibit any ordered behavior. Eisert tanglement of operations, rather than entanglement and Plenio have showed numerically that entanglement of states (Collins et al., 2001; Eisert et al., 2000b; measures do not necessarily imply the same ordering of Harrow and Shor, 2005; Linden et al., 2005). states (Eisert and Plenio, 1999). It was further shown Interestingly even two-qubit systems can reveal not analytically in (Miranowicz and Grudka, 2004) However, trivial features of entanglement decay. Namely, the it turns out that there are constraints for measures that state of the two entangled qubits, treated with unitary satisfy suitable postulates relevant in asymptotic regime. dynamics and subjected to weak noise can reach the set Namely, any such measure must lie between two extreme of separable states (see Sec. II) in finite time while coher- measures that are just two basic operational measures ence vanishes asymptotically (Zyczkowski˙ et al., 2002). distillable entanglement ED and entanglement cost EC Such an effect was investigated in detail in more realistic (Horodecki et al., 2000b). This can be seen as reflection scenarios (Dodd and Halliwell, 2004; Ficek and Tana´s, of a more general fact, that abstractly defined measures, 2006; Jak´obczyk and Jamr´oz, 2004; Lastra et al., provide bounds for operational measures which are of in- 2007; Tolkunov et al., 2005; Vaglica and Vetri, 2007; terest as they quantify how well some specific tasks can Wang et al., 2006; Yu and Eberly, 2004, 2007) (see also be performed. (Jordan et al., 2007)). It was called ”sudden death of The world of entanglement measures, even for bipar- entanglement” (Yu and Eberly, 2006) and demonstrated tite states, still exhibits puzzles. As an example may in the ingenious experiment (Almeida et al., 2007) (see serve an amazing phenomenon of locking of entanglement in this context (Santos et al., 2006)). (Horodecki et al., 2005d). For most of known bipartite Note that apart from entanglement decay a posi- measures we observe a kind of collapse: after removing a tive effects of environment was investigated: entangle- single qubit, for some states, entanglement dramatically ment generated by interference in the measurement pro- goes down. cess (Bose et al., 1999; Cabrillo et al., 1999), cavity-loss- Concerning multipartite states, some bipartite entan- induced generation of entangled atoms (Plenio et al., glement measures such as relative entropy of entangle- 1999), atom-photon entanglement conditioned by pho- ment or robustness of entanglement (Vidal and Tarrach, ton detection (Horodecki, 2001c), generation of entan- 1999) easily generalize to multipartite states. See glement from white noise (Plenio and Huelga, 2002), en- (Barnum and Linden, 2001; Eisert and Briegel, 2001) for tanglement generated by interaction with a common heat other early candidates for multipartite entanglement bath (Braun, 2002), noise-assisted preparation of entan- measures. In multipartite case a new ingredient comes gled atoms inside a cavity (Yi et al., 2003), environment in: namely, one tries to single out and quantify “truly induced entanglement in markovian dissipative dynam- multipartite” entanglement. First measure that reports ics (Benatti et al., 2003), entanglement induced by the genuinely multipartite properties is the celebrated “resid- spin chain (Yi et al., 2006). It has been demonstrated ual tangle” of (Coffman et al., 2000). It is clear, that in (Lamata et al., 2007), that it is possible to achieve an multipartite case even quantifying entanglement of pure arbitrary amount of entanglement between two atoms states is a great challenge. Interesting new schemes to by using spontaneous emitted photons, linear optics and construct measures for pure states have been proposed projective measurements. (Miyake, 2003; Verstraete et al., 2003). One of the pillars of the theory of entanglement Entanglement measures allowed for analysis of dy- is its connection with quantum cryptography, in par- namical aspects of entanglement12, including entan- ticular, with its subdomain — theory of privacy, glement decay under interaction with environment as well as a closely related classical privacy theory (Ban, 2006; Carvalho et al., 2005; Dodd and Halliwell, (Collins and Popescu, 2002; Gisin and Wolf, 1999, 2000). It seems that the most successful application of quan- tum entanglement is that it provides basic framework for quantum cryptography (despite the fact that the ba- 12 See (Amico et al., 2007; Yu and Eberly, 2005) and references sic key distribution protocol BB84 does not use entan- therein. glement directly.). This is not just a coincidence. It 8 appears, that entanglement is a quantum equivalent of 2003). Recently, the group-theory approach to entangle- what is meant by privacy. Indeed the main resource for ment has been developed in (Korbicz and Lewenstein, privacy is a secret cryptographic key: correlations shared 2006) and its connection with non-commutativity was by interested persons but not known by any other person. found. Now, in the single notion of entanglement two fundamen- In general, the structure of quantum entanglement tal features of privacy are encompassed in an ingenuous appears to be complex (see in (Gurvits and Barnum, way. If systems are in pure entangled state then at the 2002)) and many various parameters, measures, inequal- same time (i) systems are correlated and (ii) no other ities are needed to characterize its different aspects (see system is correlated with them. The interrelations be- (Alber et al., 2001a; Bengtsson and Zyczkowski,˙ 2006; tween entanglement and privacy theory are so strong, Bruß, 2002; Bruß et al., 2002; Bruß and Leuchs, 2007; that in many cases cryptographic terminology seems to Eckert et al., 2003; Terhal, 2002)). be the most accurate language to describe entanglement Finally, the list of experiments dealing with entangle- (see e.g. (Devetak and Winter, 2005)). An example of a ment13 quickly grows: ”Entanglement over long distances back-reaction — from entanglement to privacy — is the (Marcikic et al., 2004; Peng et al., 2005; Tittel et al., question of existence of bound information as a coun- 1998, 1999; Weihs et al., 1998), entanglement between terpart of bound entanglement (Gisin and Wolf, 2000). many photons (Zhao et al., 2004) and many ions In fact, the existing of such phenomenon conjectured by (Haeffner et al., 2005), entanglement of an ion and a Gisin and Wolf was founded by (Acin et al., 2004b) for photon (Blinov et al., 2004; Volz et al., 2006), entan- multipartite systems. glement of mesoscopic systems (more precisely entan- The fact that entanglement represents correlations glement between a few collective modes carried by that cannot be shared by third parties, is deeply con- many particles) ((Altewischer et al., 2002; Fasel et al., nected with monogamy - the basic feature of entangle- 2005; Julsgaard et al., 2004)), entanglement swapping ment. In 1999 Coffman, Kundu and Wootters first for- (Jennewein et al., 2002; Pan et al., 1998), the transfer of malized monogamy in quantitative terms, observing that entanglement between different carriers ((Tanzilli et al., there is inevitable trade-off between the amount of entan- 2005)), etc.” (Gisin, 2005). Let us add few recent experi- glement that a qubit A can share with a qubit B1 and ments: multiphoton path entanglement (Eisenberg et al., entanglement which same qubit A shares with some other 2005), photon entanglement from semiconductor quan- 14 qubit B2 (Coffman et al., 2000). In fact, already in 1996 tum dots (Akopian et al., 2006) . teleportation between the issue of monogamy was touched in (Bennett et al., a photonic pulse and an atomic ensemble (Sherson et al., 1996d), where it was pointed out that no system can 2006), violation of the CHSH inequality measured by be EPR correlated with two systems at the same time, two observers separated by 144 km (Ursin et al., 2006), which had direct consequences for entanglement distil- purification of two-atom entanglement (Reichle et al., lation. Monogamy expresses unshareability of entangle- 2006), increasing entanglement between Gaussian states ment (Terhal, 2004), being not only central to crypto- (Ourjoumtsev et al., 2007), creation of entangled six pho- graphic applications, but allows to shed new light to ton graph states (Lu et al., 2007). physical phenomena in many body systems such as frus- tration effect leading to highly correlated ground state (see e.g. (Dawson and Nielsen, 2004)). II. ENTANGLEMENT AS A QUANTUM PROPERTY OF The entanglement was also investigated within the COMPOUND SYSTEMS framework of special relativity and quantum field theory (Alsing and Milburn, 2002; Caban and Rembieli´nski, We accustomed to the statement that on the funda- 2005; Czachor, 1997; Jordan et al., 2006; Peres et al., mental level Nature required quantum description rather 2005; Summers and Werner, 1985; Terno, 2004) than classical one. However the full meaning of this and (see comprehensive review and references there in its all possible experimental and theoretical implications (Peres and Terno, 2004)). In particular, entangle- are far from triviality (Jozsa, 1999) In particular the “ef- ment for indistinguishable many body systems was fect” of replacement of classical concept of phase space investigated in two complementary directions. The by abstract Hilbert space makes a gap in the description first (canonical) approach bases on the tensor product of composite systems. To see this consider multipartite structure and the canonical decomposition for the states system consisting of n subsystems. According to classi- (Eckert et al., 2002; Li et al., 2001; Paskauskas and You, cal description the total state space of the system is the 2001; Schliemann et al., 2001), while the second one Cartesian product of the n subsystem spaces implying, is based on the occupation-number representation (Zanardi, 2002; Zanardi and Wang, 2002). Moreover Cirac and Verstraete (Verstraete and Cirac, 2003) 13 considered notion of entanglement in the context of su- Historically the first experimental realization of a two-qubit en- tangling quantum gate (CNOT) is due to (Monroe et al., 1995). perselection rules more generally. However it seems that 14 There was a controversy whether in previous experiment of there is still controversy concerning what is entanglement (Stevenson et al., 2006) entanglement was actually detected for indistinguishable particles (Wiseman and Vaccaro, (Lindner et al., 2006) 9
that the total state is always a product state of the n four Bell-state entangled basis separate systems. In contrast according to the quantum 1 1 formalism the total Hilbert space H is a tensor product ψ± = ( 0 1 1 0 ) φ± = ( 0 0 1 1 ). n √2 | i | i±| i | i √2 | i | i±| i | i of the subsystem spaces H = Hl. Then the super- ⊗l=1 (3) position principle allows us to write the total state of the system in the form: These states (called sometimes EPR states) have re- markable properties namely if one measures only at one of the subsystems one finds it with equal probability in ψ = cin in , (1) state 0 or state 1 . However, the result of the mea- | i i | i | i | i Xn surements for both subsystems are perfectly correlated. This is just feature which was recognized by Schr¨odinger: where i = i ,i ...i is the multiindex and i = we know nothing at all about the subsystems, although n 1 2 n | ni i1 i2 ... in , which cannot be, in general, described we have maximal knowledge of the whole system because |asi⊗| a producti ⊗| ofi states of individual subsystems15 ψ = the state is pure (see Sec. V). There is another holistic ψ ψ ψ . | i 6 feature, that unitary operation of only one of the two | 1i ⊗ | 2i⊗···⊗| ni This means, that it is in general not possible, to assign subsystems suffices to transform from any Bell states to a single state vector to any of n subsystems. It express any of the other three states. Moreover, Braunstein et formally a phenomenon of entanglement, which in con- al. showed that the Bell states are eigenstates of the Bell trast to classical superposition, allows to construct ex- operator (16) and they maximally violate Bell-CHSH in- ponentially large superposition with only linear amount equality (17) (see Sec. IV) (Braunstein et al., 1992). of physical resources. It is just what allows to perform The Bell states are special cases of bipartite maximally entangled states on Hilbert space Cd Cd given by nonclassical tasks. The states on left hand side (LHS) of ⊗ (1) appear usually as a result of direct physical interac- ψ = U U Φ+ (4) tions. However, the entanglement can be also generated | i A ⊗ B| d iAB indirectly by application of the projection postulate (en- where tanglement swapping). 1 d In practice we encounter mixed states rather than pure. Φ+ = i i (5) | d i √ | i| i Entanglement of mixed states is not longer equivalent to d i=1 being non-product, as in the case of pure states. Instead, X is the ”canonical” maximally entangled state. In this pa- one calls a mixed state of n systems entangled if it cannot per, a maximally entangled state will be also called EPR be written as a convex combination of product states16 (Einstein-Podolsky-Rosen) state or singlet state, since it (Werner, 1989a): is equivalent to the true singlet state up to local unitary transformations (for d = 2 we call it also e-bit). We will i i ρ = piρ . . . ρ . (2) also often drop the index d. 6 1 ⊗ ⊗ n i The question whether a mixture of Bell states is still X entangled, is quite nontrivial. Actually it is the case if 1 The states which are not entangled in the light of the and only if one of eigenvalues is greater than 2 (see Sec. above definition are called separable. In practice it is VI). hard to decide if a given states is separable or entan- So far the most widely used source of entangle- gled basing on the definition itself. Therefore one of the ment are entangled-photon states produced by nonlin- fundamental problems concerning entanglement is the so ear process of parametric down-conversion of type I called separability problem (see Secs. VI-X). or of type II corresponding to weather entangled pho- It should be noted in the above context that the ac- tons of the down-conversion pair are generated with tive definition of entanglement states was proposed re- the same polarization or orthogonal polarization respec- cently. Namely entangled states are the ones that can- tively. In particular using parametric down-conversion not be simulated by classical correlations (Masanes et al., one can produce Bell-state entangled basis (3). There 2007). This interpretation defines entanglement in terms are also many other sources of entangled quantum sys- of the behavior of the states rather than in terms of tems, for instance: entangled photon pairs from calcium preparation of the states. atoms (Kocher and Commins, 1967), entangled ions pre- pared in electromagnetic Paul traps (Meekhof et al., Example. for bipartite systems the Hilbert space H = 1996), entangled atoms in quantum electrodynamic cavi- H H with dimH = dimH = 2 is spanned by the 1 ⊗ 2 1 2 ties (Raimond et al., 2001), long-living entanglement be- tween macroscopic atomic ensembles (Hald et al., 1999; Julsgaard et al., 2001), entangled microwave photons from quantum dots (Emary et al., 2005), entangle- 15 Sometimes instead of notation ψ φ we use simply ψ φ and for i j even shorter ij . | i⊗| i | i| i ment between nuclear spins within a single molecule 16 Note,| i⊗| thati classical probability| i distributions can always be writ- (Chen et al., 2006a) entanglement between light and an ten as mixtures of product distributions. atomic ensembles (Muschik et al., 2006). 10
In the next section, we will present briefly pioneering (Bennett et al., 1992), because this is sufficient for pri- entanglement-based communications schemes using Bell vacy: if Alice and Bob have true EPR state, then no- entangled states. body can know results of their measurements. This is what actually happened, for a long time only this ap- proach was developed. In this case the eavesdropper Eve III. PIONEERING EFFECTS BASED ON obeys the rules of quantum mechanics. We discuss it in ENTANGLEMENT Sec. XIX. The second path was to treat EPR state as the source of strange correlations that violate Bell in- A. Quantum key distribution based on entanglement equality (see Sec. IV). This leads to new definition of security: against eavesdropper who do not have to obey The first invention of quantum information theory, the rules of quantum mechanics, but just the no-faster- which involves entanglement, was done by A. Ekert then-light communication principle. The main task of (Ekert, 1991). There were two well known facts: exis- this approach, which is an unconditionally secure proto- tence of highly correlated state17 col has been achieved only recently (Barrett et al., 2005; Masanes et al., 2006; Masanes and Winter, 2006). 1 ψ− = ( 0 1 1 0 ) (6) | i √2 | i| i − | i| i B. Quantum dense coding and Bell inequalities (violated by these states). Ekert showed that if put together, they become use- In quantum communication there holds a reasonable ful producing private cryptographic key. In this way bound on the possible miracles stemming from quan- he discovered entanglement based quantum key distri- tum formalism. This is the Holevo bound (Holevo, 1973). bution as opposed to the original BB84 scheme which Roughly speaking it states, that one qubit can carry at uses directly quantum communication. The essence of most only one bit of classical information. In 1992, Ben- the protocol is as follows: Alice and Bob can obtain nett and Wiesner have discovered a fundamental prim- from a source the EPR pairs. Measuring them in ba- itive, called dense coding, which can come around the sis 0 , 1 , Alice and Bob obtain string of perfectly Holevo bound. Dense coding allows to communicate two (anti)correlated{| i | i} bits, i.e. the key. To verify whether it is classical bits by sending one a priori entangled qubit. secure, they check Bell inequalities on selected portion of Suppose Alice wants to send one of four messages to pairs. Roughly speaking, if Eve knew the values that Al- Bob, and can send only one qubit. To communicate two ice and Bob obtain in measurement, this would mean that bits sending one qubit she needs to send a qubit in one of the values exist before the measurement, hence Bell’s in- 22 = 4 states. Moreover the states need to be mutually equalities would not be violated. Therefore, if Bell in- orthogonal, as otherwise Bob will have problems with equalities are violated, the values do not exist before Al- discriminating them, and hence the optimal bound 2 will ice and Bob measurement, so it looks like nobody can not be reached. But there are only two orthogonal states know them.18 First implementations of Ekert’s cryptog- of one qubit. Can entanglement help here? Let Alice raphy protocol has been performed using polarization- and Bob instead share a EPR state. Now the clever idea entangled photons from spontaneous parametric down- comes: it is not that qubit which is sent that should be conversion (Naik et al., 2000) and entangled photons in in four orthogonal state, but the pair of entangled qubits energy-time (Tittel et al., 2000). together. Let us check how it works. Suppose Alice and After Ekert’s idea, the research in quantum cryptog- Bob share a singlet state (6). If Alice wants to tell Bob raphy could have taken two paths. One was to treat one of the four events k 0, 1, 2, 3 she rotates her qubit violation of Bell inequality merely as a confirmation that (entangled with Bob) with∈{ a corresponding} transforma- Alice and Bob share good EPR states, as put forward in tion σk:
1 0 0 1 σ = , σ = , 0 0 1 1 1 0 17 This state is also referred to as singlet, EPR state or EPR pair. If not explicitly stated, we will further use these names to denote 1 0 0 1 σ2 = , iσ3 = − . (7) any maximally entangled state, also in higher dimensions, see 0 1 − 1 0 section VI.B.3. − 18 In fact, the argument is more subtle. This is because in principle the values which did not preexist could come to exist in a way The singlet state (6) rotated by σk on Alice’s qubit be- that is immediately available to a third party - Eve, i.e. the val- comes the corresponding ψ Bell state19. Hence ψ = | ki | ki ues that were not known to anybody could happen to be known [σk]A IB ψ0 is orthogonal to ψk′ = [σk′ ]A IB ψ0 to everybody when they come to exist. To cope with this problem ⊗ | i | i ⊗ | i Ekert has used the fact that singlet state can not be correlated with any environment. Recently it turned out that one can argue basing solely on Bell inequalities by means of so called monogamy of nonlocal correlations (Acin et al., 2006a; Barrett et al., 2005, 19 In correspondence with Bell basis defined in Eq. (3) we have + + 2006; Masanes and Winter, 2006). here ψ0 = ψ , ψ1 = φ , ψ2 = ψ , ψ3 = φ . | i | −i | i | −i | i | i | i | i 11 for k = k′ because Bell states are mutually orthogonal. essence of teleportation: a quantum state is transferred Now if6 Bob gets Alice’s half of the entangled state, after from one place to another: not copied to other place, but rotation he can discriminate between 4 Bell states, and moved to that place. But how to perform this with a pair infer k. In this way Alice sending one qubit has given of maximally entangled qubits? Bennett, Brassard, Cre- Bob log 4 = 2 bits of information. peau Jozsa, Peres and Wootters have found the answer Why does not this contradict the Holevo bound? This to this question in 1993 (Bennett et al., 1993). is because the communicated qubit was a priori entan- To perform teleportation, Alice needs to measure her gled with Bob’s qubit. This case is not covered by Holevo qubit and part of a maximally entangled state. Interest- bound, leaving place for this strange phenomenon. Note ingly, this measurement is itself entangling: it is projec- also that as a whole, two qubits have been sent: one tion onto the basis of four Bell states (3). Let us follow was needed to share the EPR state. One can also in- the situation in which she wants to communicate a qubit terpret this in the following way: sending first half of in state q = a 0 +b 1 on system A with use of a singlet | i | i | i singlet state (say it is during the night, when the channel state residing on her system A′ and Bob’s system B. The is cheaper) corresponds to sending one bit of potential total initial state which is communication. It is thus just as creating the possibility 1 of communicating 1 bit in future: at this time Alice may ψ ′ = q [ 0 0 + 1 1 ] ′ (8) AA B A √ A B not know what she will say to Bob in the future. During | i | i ⊗ 2 | i| i | i| i day, she knows what to say, but can send only one qubit can be written using the Bell basis (3) on the system AA′ (the channel is expensive). That is, she sent only one bit in the following way: of actual communication. However sending the second half of singlet as in dense coding protocol she uses both 1 + ψ ′ = [ φ ′ (a 0 + b 1 ) bits: the actual one and potential one, to communicate | AA Bi 2 | iAA | iB | iB in total 2 classical bits. Such an explanation assumes + φ− AA′ (a 0 B b 1 B) that Alice and Bob have a good quantum memory for | +i | i − | i + ψ ′ (a 1 + b 0 ) storing EPR states, which is still out of reach of current | iAA | iB | iB + ψ− ′ (a 1 b 0 )]. (9) technology. In the original dense coding protocol, Al- | iAA | iB − | iB ice and Bob share pure maximally entangled state. The Now when Alice measures her systems AA in this basis, possibility of dense coding based on partially entangled ′ she induces equiprobably the four corresponding states pure and mixed states in multiparty settings were con- on Bob’s system. The resulting states on system B are sidered in (Barenco and Ekert, 1995; Bose et al., 2000; very similar to the state of qubit q which Alice wanted Bruß et al., 2005; Hausladen et al., 1996; Mozes et al., to send him. They however admix| i to the initial state 2005; Ziman and Buzek, 2003) (see Sec. XIV.F. The of system B. Thus Bob does not get any information first experimental implementation of quantum dense cod- instantaneously. However the output structure revealed ing was performed in Innsbruck (Mattle et al., 1996a), in the above equation can be used: now Alice tells to Bob using as a source of polarization-entangled photons (see her result via telephone. Accordingly to those two bits of further experiments with using nuclear magnetic reso- information (which of Bell states occurred on AA ) Bob nance (Fang et al., 1999), two-mode squeezed vacuum ′ rotates his qubit by one of the four Pauli transformations state (Mizuno et al., 2005) and controlled dense cod- (7). This is almost the end. After each rotation, Bob gets ing with EPR state for continuous variable (Jietai et al., q at his site. At the same time, Alice has just one of 2003)). | i the Bell states: the systems A and A′ becomes entangled after measurement, and no information about the state q is left with her. That is, the no cloning principle is C. Quantum teleportation |observed,i while the state q was transferred to Bob. There is a much simpler| i way to send qubit to Bob: Suppose Alice wants to communicate to Bob an un- Alice could just send it directly. Then however, she has known quantum bit. They have at disposal only a clas- to use a quantum channel, just at the time she wants to sical telephone, and one pair of entangled qubits. One transmit the qubit. With teleportation, she might have way for Alice would be to measure the qubit, guess the send half of EPR pair at any earlier time, so that only state based on outcomes of measurement and describe it classical communication is needed later. to Bob via telephone. However, in this way, the state This is how quantum teleportation works in theory. will be transferred with very poor fidelity. In general an This idea has been developed also for other communi- unknown qubit can not be described by classical means, cation scenarios (see (D¨ur and Cirac, 2000c)). It be- as it would become cloneable, which would violate the came immediately an essential ingredient of many quan- main principle of quantum information theory: a qubit tum communication protocols. After pioneering exper- in an unknown quantum state cannot be cloned (Dieks, iments (Boschi et al., 1998; Bouwmeester et al., 1997; 1982; Wootters and Zurek, 1982). Furusawa et al., 1998), there were beautiful experiments However, Alice can send the qubit to Bob at the price performing teleportation in different scenarios during last of simultaneously erasing it at her site. This is the decade (see e.g. (Barrett et al., 2004; Marcikic et al., 12
2004; Nielsen et al., 1998; Riebe et al., 2004; Ursin et al., bits needed in order to achieve this task is called the 2004)). For the most recent one with mesoscopic objects, communication complexity of the function f. see (Sherson et al., 2006). Again, one can ask if entanglement can help in this case. This question was first asked by Cleve and Buhrman (Cleve and Buhrman, 1997) and in a differ- D. Entanglement swapping ent context independently by Grover (Grover, 1997), who showed the advantage of entanglement assisted dis- Usually quantum entanglement originates in certain tributed computation over the classical one. direct interaction between two particles placed close to- Consider the following example which is a three-party gether. Is it possible to get entangled (correlated in quan- version of the same problem (Buhrman et al., 2001a). tum way) some two particles which have never interacted Alice Bob and Clare get two bits each: (a1,a0) (b1,b0) in the past? The answer is positive (Bennett et al., 1993; and (c1,c0) representing a binary two-digit numbers a = Yurke and Stoler, 1992b; Zukowski˙ et al., 1993). a1a0, b = b1b0 and c = c1c0. They are promised to have Let Alice share a maximally entangled state φ+ = 1 | i ( 00 + 11 ) with Clare, and Bob share the same a0 b0 c0 =0. (11) √2 | i | i AC ⊕ ⊕ state with David: The function which they are to compute is given by + + φ AC φ BD. (10) | i ⊗ | i f(a,b,c)= a b c (a b c ). (12) 1 ⊕ 1 ⊕ 1 ⊕ 0 ∨ 0 ∨ 0 Such state can be obviously designed in such a way, that particles A and D have never seen each other. Now, Clare It is easy to see, that announcing of four bits is sufficient and Bob perform a joint measurement in Bell basis. It so that all three parties could compute f. One party turns out that for any outcome, the particles A and D announces both bits (say it is Alice) a1a0. Now if a0 = 1, collapse to some Bell state. If Alice and Bob will get then the other parties announces their first bits b1 and to know the outcome, they can perform local rotation, to c1. If a0 = 0, then one of the other parties, (say Bob) + obtain entangled state Φ . In this way particles of Alice announces b1 b0 while Clare announces c1. In both cases AD ⊕ and David are entangled although they never interacted all parties compute the function by adding announced directly with each other, as they originated from different bits modulo 2. Thus four bits are enough. A bit more sources. tricky is to show that four bits are necessary, so that the One sees that this is equivalent to teleporting one of classical communication complexity of the above function the EPR pair through the second one. Since the protocol equals four (Buhrman et al., 2001a). is symmetric, any of the pairs can be chosen to be either Suppose now, that at the beginning of the protocol the channel, or the teleported pair. Alice Bob and Clare share a quantum three-partite en- This idea has been adopted in order to perform quan- tangled state: tum repeaters (D¨ur et al., 1999a), to allow for distribut- ing entanglement — in principle — between arbitrarily 1 ψABC = [ 000 011 101 110 ]ABC, (13) distant parties. It was generalized to a multipartite sce- | i 2 | i − | i − | i − | i nario in (Bose et al., 1998). Swapping can be used as a tool in multipartite state distribution, which is for exam- such that each party holds a corresponding qubit. It is ple useful in quantum cryptography. enough to consider the action of Alice as the other parties The conditions which should be met in optical im- will do the same with their classical and quantum data plementation of entanglement swapping (as well as tele- respectively. ˙ Alice checks her second bit. If a0 = 1 she does a portation) have been derived in (Zukowski et al., 1993). 20 Along those lines entanglement swapping was realized in Hadamard on her qubit. Then she measures it in lab (Pan et al., 1998). 0 , 1 basis noting the result rA. She then announces {|a biti |ai} r . 1 ⊕ A One can check, that if all three parties do the same, E. Beating classical communication complexity bounds the r bits with “quantum origin” fulfills rA rB rC = a b− c . It gives in turn ⊕ ⊕ with entanglement 0 ∨ 0 ∨ 0 (a r ) (b r ) (c r )= Yao (Yao, 1979) has asked the following question: how 1 ⊕ A ⊕ 1 ⊕ B ⊕ 1 ⊕ C much communication is needed in order to solve a given a1 b1 c1 (rA rB rC )= ⊕ ⊕ ⊕ ⊕ ⊕ problem distributed among specially separated comput- a1 b1 c1 (a0 b0 c0)= f(a,b,c). (14) ers? To be more concrete, one can imagine Alice having ⊕ ⊕ ⊕ ∨ ∨ got the n-bit string x and Bob having got n-bit string y. Their task is to infer the value of some a priori given function f(x, y) taking value in 0, 1 , so that finally 20 { } Hadamard transformation is a unitary transformation of basis both parties know the value. The minimal amount of which changes 0 into 1 ( 0 + 1 ) and 1 into 1 ( 0 1 ). | i √2 | i | i | i √2 | i−| i 13
Thus three bits are enough. The fourth, controlled by cannot be generated from classical correlations. As a common quantum entanglement, got hidden in the three matter of fact, Bell in his proof assumed perfect corre- others, hence need not be announced. lations exhibited by the singlet state. However, in real One can bother that the quantum state (13) should en- experiments such correlations are practically impossible. ter the communication score, as it needs to be distributed Inspired by Bell’s paper, Clauser, Horne, Shimony and among the parties. Thus in addition to 3 bits they need Holt (CHSH) (Clauser et al., 1969) derived a correlation to send qubits. However similarly as in case of dense cod- inequality, which provides a way of experimental testing ing, the entangled state is independent of the function, of the LHVM as independent hypothesis separated from i.e. it can be prepared prior to the knowledge of the form quantum formalism. Consider correlation experiment in of the function. which the variables (A1, A2) are measured on the one It is known, that if exact evaluation of the function is of the subsystems of the whole system while (B1,B2) on required (as in the described case), sharing additionally the other one, and that the subsystems are spatially sepa- correlated bit-strings (shared randomness) by the par- rated. Then the LHVM imposes the following constraints ties can not help them. Thus we see, that entanglement on the statistics of the measurements on the sufficiently which can serve itself as a source of shared randomness, large ensemble of systems21 is a stronger resource. If one allows nonzero probability E(A ,B )+ E(A ,B )+ E(A ,B ) E(A ,B ) 2, of error, then classical correlations can help, but it was | 1 1 1 2 2 1 − 2 2 | ≤ shown in a bipartite scenario, that entanglement allows (15) for smaller error. (Buhrman et al., 2001a). Entangle- where E(Ai,Bj ) is the expectation value of the correla- ment can beat classical correlations even exponentially tion experiment AiBj . as shown by R. Raz (Raz, 1999). Other schemes with This is the CHSH inequality that gives a bound on exponential advantage are provided in (Gavinsky et al., any LHVM. It involves only two-partite correlation func- 2006b). They are closely related to the phenomenon tion for two alternative dichotomic measurements and it called quantum fingerprinting (Buhrman et al., 2001b; is complete in a sense that if full set of such inequali- Gavinsky et al., 2006a). ties is satisfied there exists a joint probability distribu- Interestingly, though the effect is of practical impor- tion for the outcomes of the four observables, which re- tance, in its roots there is purely philosophical question turns the measured correlation probabilities as marginals of type: “Is the Moon there, if nobody looks?” (Mermin, (Fine, 1982). 1985). Namely, if the outcomes of the Alice, Bob and In quantum case the variables convert into operators Clare measurements would exist prior to measurement, and one can introduce the CHSH operator then they could not lead to reduction of complexity, be- = A (B + B )+ A (B B ), (16) cause, the parties could have these results written on BCHSH 1 ⊗ 1 2 2 ⊗ 1 − 2 paper and they would offer three bit strategy, which is A a1 A a2 B not possible. As a matter of fact, the discoveries of re- where 1 = σ, 2 = σ, similarly for 1 and B · · duction of communication complexity used GHZ paradox 2, σ = (σx, σy, σz) is the vector of Pauli operators, a in Mermin version, which just says that the outcomes of = (ax,ay,az) etc., are unit vectors describing the mea- the four possible measurements given by values of a , b surements that the parties: A (Alice) and B (Bob) per- 0 0 form. Then, the CHSH inequality requires that the con- and c0 (recall the constraint a0 b0 c0) performed on the state (13) cannot preexist. ⊕ ⊕ dition In (Brukner et al., 2004) it was shown that this is quite Tr( ρ) 2 (17) generic: for any correlation Bell inequality for n systems, | BCHSH |≤ a state violating the inequality allows to reduce commu- is satisfied for all states ρ admitting a LHVM. nication complexity of some problem. For recent experi- Quantum formalism predicts Tsirelson inequality ment see (Trojek et al., 2005). (Tsirelson, 1980)
CHSH QM = Tr( CHSH ρ) 2√2 (18) IV. CORRELATION MANIFESTATIONS OF |hB i | | B |≤ ENTANGLEMENT: BELL INEQUALITIES. for all states ρ and all observables A1, A2, B1, B2. Clearly, the CHSH inequality can be violated for some A. Bell theorem: CHSH inequality. choices of observables, implying violation of LHVM. For singlet state ρ = ψ− ψ− , there is maximal violation | ih | The physical consequences of the existence of entan- Tr( CHSH ρ) = 2√2 which corresponds to Tsirelson gled (inseparable) states are continuously subject of in- bound.| B | tensive investigations in the context of both EPR paradox and quantum information theory. They manifest itself, in particular, in correlation experiments via Bell theo- 21 rem which states that the probabilities for the outcomes It is assumed here that the variables Ai and Bi for i = 1, 2 are obtained when suitably measuring some quantum states dichotomic i.e. have values 1. ± 14
B. The optimal CHSH inequality for 2 × 2 systems C. Nonlocality of quantum states and LHV model
1. Pure states At the beginning of 90’s there were two basic questions: Firstly, it was hard to say whether a given state violates The second more fundamental question was: Are there the CHSH inequality, because one has to construct a re- many quantum states, which do not admit LHV model? spective Bell observable for it. In addition, given a mixed more precisely: which quantum states do not admit LHV state, there was no way to ensure whether ii satisfies the model? Even for pure states the problem is not com- CHSH inequality for each Bell observable. This problem pletely solved. was solved completely for an arbitrary quantum states ρ Gisin proved that for the standard (ie. nonsequential) of two qubits by using Hilbert-Schmidt space approach projective measurements the only pure 2-partite states (Horodecki et al., 1995). Namely, n-qubit state can be which do not violate correlation CHSH inequality (15) are written as product states and they are obviously local (Gisin, 1991; Gisin and Peres, 1992). Then Popescu and Rohrlich 3 showed that any n-partite pure entangled state can al- 1 ρ = t σ1 σn , (19) ways be projected onto a 2-partite pure entangled state 2n i1...in i1 ⊗···⊗ in i1...in=0 by projecting n 2 parties onto appropriate local pure X states (Popescu and− Rohrlich, 1992). Clearly, it needs an additional manipulations (postselection). Still the prob- k where σ0 is the identity operator in the Hilbert space of lem whether Gisin theorem, can be generalized without k postselection for an arbitrary n-partite pure entangled qubit k, and the σik correspond to the Pauli operators for three orthogonal directions ik = 1, 2, 3. The set of states, remains open. In the case of three parties there real coefficients t = Tr[ρ(σ σ )] forms a are generalized GHZ states (Scarani and Gisin, 2001b; i1...in i1 ⊗···⊗ in correlation tensor Tρ. In particular for the two-qubit Zukowski˙ et al., 2002) that do not violate the MABK in- system the 3 3 dimensional tensor is given by t := equalities (Ardehali, 1992; Belinskii and Klyshko, 1993; × ij Tr[ρ(σi σj )] for i, j =1, 2, 3. Mermin, 1990a) (see next subsec.). More generally it has ⊗ ˙ In this case one can compute the mean value of an been shown (Zukowski et al., 2002) that these states do not violate any Bell inequality for n-partite correlation arbitrary CHSH in an arbitrary fixed state ρ and then B functions for experiments involving two dichotomic ob- maximize it with respect to all CHSH observables. In B servables per site. Acin et al. and Chen et al. considered result we have: max CHSH Tr( CHSH ρ) = 2 M(ρ) = B | B | a Bell inequality that shows numerical evidence that all 2 t2 + t2 , where t2 and t2 are the two largest eigen- 11 22 11 22 p three-partite pure entangled state violate it (Acin et al., values of T T T , T T is the transposed of T . p ρ ρ ρ ρ 2004a; Chen et al., 2004). Recently a stronger Bell in- It follows that for 2 2 systems the necessary and suf- equalities with more measurement settings was presented ficient criterion for the× violation of the CHSH inequality (Laskowski et al., 2004; X.-H.Wu and Zong, 2003) which can be written in the form can be violated by a wide class of states, including the generalized GHZ states (see also (Chen et al., 2006b)).
M(ρ) > 1. (20) 2. Mixed states
The quantity M(ρ) depends only on the state parame- In the case of noisy entangled states the problem ap- ters and contains all the information that is needed to peared to be much more complex. A natural conjecture decide whether a state violates a CHSH inequality. Note was that only separable mixed states of form (38) ad- that using M(ρ) one can define a measure of violation mit a LHV model. Surprisingly, Werner constructed one parameter of family of U U invariant states (see Sec. of the CHSH inequality B(ρ) = max 0, M(ρ) 1 ⊗ { − } VI.B.9) in d d dimensions, where U is a unitary op- (Miranowicz, 2004b), which for anq arbitraryp two-qubit erator and showed× that some of them can be simulated pure state equals to measures of entanglement: negativity by such model (Werner, 1989a). In particular, two-qubit and concurrence (see Sec. XV.C.2.b). Clearly, in general, (d = 2) Werner states are mixtures of the singlet ψ− both M(ρ) and B(ρ) are not measures of entanglement with white noise of the form | i as they do not behave monotonously under LOCC. How- I ever, they can be treated as entanglement parameters, ρ = p ψ− ψ− + (1 p) . (21) in a sense that they characterize a degree of entangle- | ih | − 4 ment. In particular the inequality (20) provides practical It has been shown, using the criterion (20) tool for the investigation of nonlocality of the arbitrary (Horodecki et al., 1995), that the CHSH inequality 1 two qubits mixed states in different quantum-information is violated when 2− 2
2 is not maximally entangled. the n-qubit quantum correlation function has the form: Interestingly, it has been recognized that there is a tradeoff between the local commutativity and Bell in- E (k) = Tr[ρ n a (k ) ~σ]= q ⊗j=1 j j · equality non-violating (Roy, 2005). This idea was devel- T, a ~a . (24) h 1 ⊗···⊗ ni oped in (Seevinck and Uffink, 2007). Note that there are Bell inequalities of other type Inserting Eq(k) to the expression (23) and using (19) such as: Bell inequalities in phase space (Auberson et al., one obtains that the correlations between the measure- 2003); and entropic Bell inequalities, which involve clas- ment on n qubits admit an LHVM iff in any set of sical entropy (Braunstein and Caves, 1988). However, local coordinate system and for any set of unit vec- the entropic ones are weaker than the CHSH inequal- c j j 2 tors j = (c1,c2) R the following inequality holds ity. In (Nagata et al., 2004) Bell inequalities were con- ˙ ∈ (Zukowski and Brukner, 2002) sidered in the context of rotational invariance. It should
2 be noted, that the pioneering investigation of the Bell inequalities in the relativistic context was performed by T mod c1 ...cn t 1, (25) c1...cn ≡ i1 in | i1...in |≤ Werner and Summers (Summers and Werner, 1985) and i1...i Xn Czachor (Czachor, 1997). mod where Tc1...cn is a component of moduli of correlation tensor T , along directions defined by the unit vectors cj . In the spirit of the Peres separability criterion one can E. Logical versions of Bell’s theorem express the above condition as follows: within LHVM T mod is also a possible correlation tensor. The violation of Bell inequalities as a paradigmatic test Note that the Bell-WWZB operator for n-partite sys- of quantum formalism has some unsatisfactory feature as tem with two dichotomic observables per site admits it only applies to a statistical measurement procedure. It spectral decomposition into n-qubit GHZ states (82) is intriguing, that the quantum formalism via quantum which lead to maximal violations (Scarani and Gisin, entanglement offers even stronger departure from clas- 2001b), (Werner and Wolf, 2001c). Thus the GHZ states sical intuition based on the logical argument which dis- play a similar role for the WWZB inequality as the Bell cusses only perfectly correlated states. This kind of argu- states for the CHSH one. ment was discovered by Greenberger, Horne and Zeilinger The WWZB inequalities are important tool for the (GHZ) and applies for all multipartite systems that are in the same GHZ state i.e. ψ = 1 ( 000 + 111 ) . investigation of possible connections between quantum | iABC √2 | i | i ABC nonlocality distillability and entanglement for n-qubit The authors showed that any deterministic LHVM pre- systems. In particular, it has been shown that the vi- dicts that a certain outcome always happen while quan- olation of the WWZB inequality by a multi-qubit states tum formalism predicts it never happens. implies that pure entanglement can be distilled from it. The original GHZ argument for qubits has been But the protocol may require that some of the parties subsequently developed (Cabello, 2001a,b; Cerf et al., join into several groups (Acin, 2002; Acin et al., 2003c). 2002a; Chen et al., 2003; Greenberger et al., 2005a,b; 17
Hardy, 1993; Mermin, 1990b) and extended to contin- v(zAzA′ )v(xAxA′ )v(zBxB′ )v(xB zB′ )= 1. uous variables (Chen and Zhang, 2001; Clifton, 2000; − Massar and Pironio, 2001) and it is known as the “all- Multiplying the above equations except the last one, one versus-nothing” proof of Bell theorem or “Bell theorem gets v(zAzA′ )v(xAxA′ )v(zB xB′ )v(xB zB′ ) = 1, that contra- without inequalities”. The proofs are purely logical. dicts the last one. However, in real experiments ideal measurements and In real experiments the nonlocality tests are based on perfect correlations are practically impossible. To over- the estimation of the Bell type operator O = z z − A · B − come the problem of a “null experiment” Bell-type in- z′ z′ x x x′ x′ + z z′ z z′ + x x′ A · B − A · B − A · B A A · B · B A A · equalities are needed. x x′ + z x′ z x′ + x z′ x z′ z z′ x x′ B · B A · A · B B A · A · B B − A A · A A · An important step in this direction was the re- zBxB′ xB zB′ and Bell-Mermin type inequality (Cabello, duction of the GHZ proof to the two-particle non- 2001a,b;· Chen et al., 2003) maximally (Hardy, 1993) and maximally entangled systems of high-dimensionality (Chen et al., 2005d; O L 7. (27) Durt et al., 2001; Kaszlikowski et al., 2000; Pan et al., h i ≤ 2001; Torgerson et al., 1995). Recently a two-particle all- It is easy to check that the operator O satisfies the versus-nothing nonlocality tests were performed by using relation O Ψ = 9 Ψ , which is in contradiction with two-photon so-called hyperentanglement (Cinelli et al., the above inequality| i | impliedi by LHVM. In a recent ex- 2005; Yang et al., 2005b). It is instructive to see how the periment (Yang et al., 2005b) observed value for O was GHZ argument works in this special case, when a single 8.56904 0.00533. It demonstrates a contradiction of source produces a two-photon state simultaneously max- quantum± predictions with LHVM (see in this context re- imally entangled both in the polarization and in the path lated experiment (Cinelli et al., 2005)). A novel nonlo- degrees of freedom. It can be written in the form: cality test “stronger two observer all versus nothing” test (Cabello, 2005) have been performed by use of a 4-qubit 1 Ψ = ( H V V H )( R L L R ). linear cluster state via two photons entangled both in po- | i 2 | iA| iB −| iA| iB | iA| iB −| iA| iB (26) larization and linear momentum (Vallone et al., 2007). Here photon-A and photon-B pertain to spatially sep- Note that, the GHZ argument has been extended to the arated observers Alice and Bob, H and V stand for entangled two-particle states that are produced from two photons with horizontal and vertical| i polarization| i and R independent sources (Yurke and Stoler, 1992b) by the ˙ , L are the spatial (path) modes. process of “entanglement swapping” (Zukowski et al., Now, one can define the set of the nine local observ- 1993). The complete proof depends crucially on the inde- ables corresponding to Ψ to be measured by Alice and pendence of the two sources and involves the factorization | i Bob as follows: a = (z′, x , x z′), b = (z , x′ ,z x′ ), of the Bell function (Greenberger et al., 2005a). i i i i · i i i i i · i c = (z z′, x x′ ,z z′ x x′ ), where i = A, B and z = σ , i i i i i i i · i i i zi xi = σxi , zi′ = σz′ i , xi′ = σx′ i are Pauli-type operators for the polarization and the path degree of freedom respec- F. Violation of Bell inequalities: general remarks tively. The observables are arranged in three groups a, b, and c, where the observables of each group are measured There is a huge literature concerning interpretation by one and the same apparatus. of the “Bell effect”. The most evident conclusion from Then quantum formalism makes the following predic- those experiments is that it is not possible to construct tions for perfect correlations: a LHVM simulating all correlations observed for quan- tum states of composite systems. But such conclusion z z Ψ = Ψ ,z′ z′ Ψ = Ψ , A · B | i − | i A · B | i − | i is not surprising. What is crucial in the context of the xA xB Ψ = Ψ , x′ x′ Ψ = Ψ , Bell theorem is just a gap between quantum and classical · | i − | i A · B | i − | i description of the correlations, which gets out of hand. zAzA′ zBzB′ Ψ = Ψ , xAxA′ xBxB′ Ψ = Ψ , · | i | i · | i | i Nature on its fundamental level offers us a new kind z x′ z x′ Ψ = Ψ , x z′ x z′ Ψ = Ψ , A · A · B B | i | i A · A · B B | i | i of statistical non-message-bearing correlations, which are z z′ x x′ z x′ x z′ Ψ = Ψ . A A · A A · B B · B B | i − | i encoded in the quantum description of states of com- 22 According to EPR the perfect correlations allow assign pound systems via entanglement. They are “nonlocal” the pre-existing measurement values v(X) to the above in the sense, that observables X. Then the consistency between quantum i) They cannot be described by a LHVM. predictions and any LHVM gives the following relations ii) They are nonsignaling as the local measurements between the “elements of reality”: performed on spatially separated systems cannot be used to transmit messages. v(z )v(z )= 1, v(z′ )v(z′ )= 1, A B − A B − v(x )v(z )= 1, v(x′ )v(x′ )= 1, A B − A B − v(z z′ )v(z )v(z′ )=1, v(x x′ )v(x )v(x′ )=1, A A B B A A B B 22 The term nonlocality is somewhat misleading. In fact there is a v(zA)v(xA′ )v(zBxB′ )=1, v(xA)v(zA′ )v(xB zB′ )=1, breaking of conjunction of locality and counterfactuality. 18
Generally speaking, quantum compound systems can the entropy of a subsystem can be greater than the en- reveal holistic nonsignaling effects even if their subsys- tropy of the total system only when the state is entan- tems are spatially separated by macroscopic distances. gled (Horodecki and Horodecki, 1994). In other words In this sense quantum formalism offers holistic descrip- the subsystems of entangled system may exhibit more tion of Nature (Primas, 1983), where in a non-trivial way disorder than the system as a whole. In the classical the system is more than a combination of its subsystems. world it never happens. Indeed, the Shannon entropy It is intriguing that entanglement does not exhaust H(X) of a single random variable is never larger than full potential of nonlocality under the constraints of the entropy of two variables no-signaling. Indeed, it does not violate Tsirelson’s bound (18) for CHSH inequalities. On the other hand, H(X, Y ) H(X), H(X, Y ) H(Y ). (29) ≥ ≥ one can design family of probability distributions (see (Gisin, 2005) and ref. therein), which would violate It has been proved (Horodecki and Horodecki, this bound but still do not allow for signaling. They 1996; Horodecki et al., 1996c; Terhal, 2002; are called Popescu-Rohrlich nonlocal boxes, and repre- Vollbrecht and Wolf, 2002b) that analogous α-entropy sent extremal nonlocality admissible without signaling inequalities hold for separable states (Popescu and Rohrlich, 1994). We see therefore that S (ρ ) S (ρ ), S (ρ ) S (ρ ), (30) quantum entanglement is situated in an intermediate α AB ≥ α A α AB ≥ α B level between locality and maximal non-signaling non- 1 α locality. Needless to say, the Bell inequalities still in- where Sα(ρ)=(1 α)− log Trρ is the α-Renyi entropy for α 0, here ρ −= Tr (ρ ) and similarly for ρ . If volve many fascinating open problems interesting for ≥ A B AB B both philosophers and physicists (see (Gisin, 2007)). α tends to 1 decreasing, one obtains the von Neumann entropy S1(ρ) S(ρ) as a limiting case. The above inequalities≡ represent scalar separability cri- teria. For pure states these inequalities are violated if V. ENTROPIC MANIFESTATIONS OF ENTANGLEMENT and only if the state is entangled. As separable states admit LHVM one can expect, that the violation of the A. Entropic inequalities: classical versus quantum order entropic inequalities is a signature of some nonclassical features of a compound quantum system resulting from It was mentioned in the introduction that Schr¨odinger its entanglement. The above inequalities involve a non- first pointed out, that entanglement does not manifest linear functionals of a state ρ, so they can be interpreted itself only as correlations. In fact he has recognized an as a scalar separability criteria based on a nonlinear en- other aspect of entanglement, which involves profoundly tanglement witness (see Sec. VI). nonclassical relation between the information which an The α-entropy inequalities were considered in the entangled state gives us about the whole system and the context of Bell inequalities (Horodecki and Horodecki, information which it gives us about subsystems. 1996) (and quantum communication for α = 1 This new “nonintuitive” property of compound quan- (Cerf and Adami, 1997)). For α = 2 one get the inequal- tum systems, intimately connected with entanglement, ities based on the 2-Renyi entropy, which are stronger was a long-standing puzzle from both a physical and a detectors of entanglement than all Bell-CHSH inequal- mathematical point of view. The main difficulty was, ities (Horodecki, 1996; Horodecki et al., 1996c; Santos, that in contrast to the concept of correlation, which has a 2004). The above criterion was expressed in terms of the clear operational meaning, the concept of information in Tsallis entropy (Abe and Rajagopal, 2001; Tsallis et al., quantum theory was obscure till 1995, when Schumacher 2001; Vidiella-Barranco, 1999). showed that the von Neumann entropy Another interesting family of separability of non- linear criteria was derived (G¨uhne and Lewenstein, S(ρ)= Trρ log ρ (28) − 2004a) in terms of entropic uncertainty relations (Bia lynicki-Birula and Mycielski, 1975; Deutsch, 1983; has the operational interpretation of the number of qubits Kraus, 1987; Maassen and Uffink, 1988)23 using Tsallis needed to transmit quantum states emitted by a statis- entropy. In contrast to linear functionals, such as cor- tical source (Schumacher, 1995b). The von Neumann relation functions, the entropies, being a nonlinear func- entropy can be viewed as the quantum counterpart of tionals of a state ρ, are not directly measurable quanti- the Shannon entropy H(X)= p log p , p = 1, i i i i i ties. Therefore, the problem of experimental verification which is defined operationally− as the minimum number of the violation of the entropic inequalities was difficult of bits needed to communicate aP message producedP by a from both a conceptual and a technical point of view. classical statistical source associated to a random vari- able X. Fortunately in 1994 Schr¨odinger’s observation, that an entangled state gives us more information about the to- 23 Quite recently the quantum entropic uncertainty relations has tal system than about subsystems was, quantified just been expressed in the form of inequalities involving the Renyi by means of von Neumann entropy. It was shown that entropies (Bialynicki-Birula, 2006). 19
However, recently Bovino et al. demonstrated experi- R R mentally a violation of 2-entropic inequalities based on the Renyi entropy (Bovino et al., 2005). This also was first direct experimental measurement of a nonlinear en- state merging tanglement witness (see Sec. VIII.C). |Ψi |Ψi
B. 1-entropic inequalities and negativity of information A B A B′ B Entropic inequalities (30) can be viewed as a scalar EPR pairs separability criteria. This role is analogous to Bell in- equalities as entanglement witnesses. In this context a FIG. 1 The concept of state merging: before and after. natural question arises: is this all we should expect from violation of entropic inequalities? Surprisingly enough, entropic inequalities are only the tip of the iceberg which 1971 Slepian and Wolf considered the following prob- reveals dramatic differences between classical and quan- lem: how many bits does the sender (Alice) need to send tum communication due to quantum entanglement. To to transmit a message from the source, provided the re- see it consider again the entropic inequalities based on ceiver (Bob) already has some prior information about the von Neumann entropy (α = 1) which hold for sepa- the source. The amount of bits is called the partial in- rable states. They may be equivalently expressed as formation. Slepian and Wolf proved (Slepian and Wolf, 1971), that the partial information is equal to the condi- S(ρ ) S(ρ ) 0, S(ρ ) S(ρ ) 0 (31) AB − B ≥ AB − A ≥ tional entropy (33),which is always positive: H(X Y ) 0. | ≥ and interpreted as the constraints imposed on the cor- In the quantum state merging scenario, an unknown relations of the bipartite system by positivity of some quantum state is distributed to spatially separated ob- function servers Alice and Bob. The question is: how much quan- tum communication is needed to transfer Alice’s part of S(A B)= S(ρAB) S(ρB) (32) | − the state to Bob in such a way that finally Bob has total and similarly for S(B A). Clearly, the entropy of the state (Fig. 1). This communication measures the partial | subsystem S(ρA) can be greater than the total entropy information which Bob needs conditioned on its prior in- S(ρAB) of the system only when the state is entan- formation S(B). Surprisingly, Horodecki at al. proved gled. However, there are entangled states which do that necessary and sufficient number of qubits is given not exhibit this exotic property, i.e. they satisfy the by the formula (32), even if this quantity is negative. constraints. Thus the physical meaning of the func- Remarkably, this phenomenon asunders into two levels: tion S(A B) (S(B A)) and its peculiar behavior were an “classical” and quantum depend on sign of the partial enigma for| physicists.| information S(A B): | There was an attempt (Cerf and Adami, 1997) to over- 1) partial information S(A B) is positive (the inequal- come this difficulty by replacing Shannon entropies with ities (31) are not violated):| the optimal state merging von Neumann one in the formula for classical conditional protocol requires sending r = S(A B) qubits. ∼ | entropy 2) partial information S(A B) is negative (the inequal- ities (31) are violated): the| optimal state merging does H(X Y )= H(X, Y ) H(X) (33) | − not need sending qubits; in addition Alice and Bob gain r ∼= S(A B) pairs of qubits in a maximally entangled to get quantum conditional entropy in form (32) (Wehrl, state.− | 1978). Unfortunately, such approach is inconsistent with It is remarkable that we have clear division into two the concept of the channel capacity defined via the regimes: when the 1-entropy inequalities are not violated, function S(A B) called coherent information (see Sec. in the process of quantum state merging consumed and XII.A) (Schumacher− | and Nielsen, 1996)24. Moreover, so we have a nice analogy to the classical Slepian-Wolf the- defined conditional entropy can be negative. orem where partial information (equal to conditional en- Recently, the solution of the problem was pre- tropy (33)) is always positive; when the inequalities are sented within the quantum counterpart of the classical violated, apart from state merging we get additional en- Slepian-Wolf theorem called “quantum state merging” tanglement. Thus the quantum and classical regimes are ((Horodecki et al., 2005h), (Horodecki et al., 2006e)). In determined by the relations between the knowledge about the system as a whole and about its subsystem, as con- sidered by Schr¨odinger.
24 The term “coherent information” was originally defined to be a Finally, let us note that the early recognized mani- function of a state and a channel, but further its use has been festations of entanglement: (nonlocality (EPR, Bell) and extended to apply to a bipartite state. what we can call insubordination (Schr¨odinger)) had been 20 seemingly academic issues, of merely philosophical rele- A B is called separable (entangled) iff it can be (can vance. What is perhaps the most surprising twist, is Hnot⊗H be) written as a product of two vectors corresponding that both the above features qualify entanglement as a to Hilbert spaces of subsystems: resource to perform some concrete tasks. Indeed, the violation of the Bell inequalities determines ΨAB = φA ψB . (35) | i | i| i usefulness of quantum states for the sake of specific non- classical tasks, such as e.g. reduction of communication In general if the vector ΨAB is written in any orthonor- mal product basis ei ej as follows25: complexity, or quantum cryptography (see Sec.III). Sim- {| Ai ⊗ | Bi} ilarly, the violation of the entropic inequalities based on dA 1 dB 1 the von Neumann entropy (31) determines the usefulness − − Ψ = AΨ ei ej , (36) of states as a potential for quantum communication. It | ABi ij | Ai ⊗ | Bi i=0 j=0 is in agreement with the earlier results, that the negative X X value of the function S(A B) is connected with the ability | then it is product if and only if the matrix of coefficients of the system to perform teleportation (Cerf and Adami, AΨ = AΨ is of rank one. In general the rank r(Ψ) 1997; Horodecki and Horodecki, 1996) as well as with { ij } ≤ k min[dA, dB ] of this matrix is called Schmidt rank nonzero capacity of a quantum channel (Devetak, 2003; of≡ vector Ψ and it is equal to either of ranks of the Lloyd, 1997; Schumacher and Nielsen, 1996). Ψ Ψ reduced density matrices ̺A = TrB ΨAB ΨAB , ̺B = Ψ | Ψ ihΨ | Ψ TrA ΨAB ΨAB (which satisfy ̺A = A (A )† and ̺A = Ψ| ΨihT | 26 [(A )†A ] respectively ). In particular there always C. Majorization relations i j exists such a product bi-orthonormal basis e˜A e˜B in which the vector takes the Schmidt decomposition:{| i ⊗ | i} In 2001 Nielsen and Kempe discovered a stronger version of the “classical versus quantum order” r(Ψ) (Nielsen and Kempe, 2001), which connects the ma- i i ΨAB = ai e˜ e˜ , (37) jorization concept and entanglement. Namely they | i | Ai ⊗ | Bi i=0 proved, that if a state is separable than the following X inequalities where the strictly positive numbers ai = √pi correspond to the nonzero singular eigenvalues{ } λ(ρ) λ(ρ ) λ(ρ) λ(ρ ) (34) Ψ ≺ A ≺ B (Nielsen and Chuang, 2000) of A , and pi are the nonzero elements of the spectrum of either of the has to be fulfilled. Here λ(ρ) is a vector of eigenval- reduced density matrices. ues of ρ; λ(ρ ) and λ(ρ ) are defined similarly. The A B Quantum entanglement is in general both quantita- relation x y between d-dimensional vectors x and y ≺ k k tively and qualitatively considered to be a property in- (x is majorized by y) means that i=1 xi↓ i=1 yi↓, variant under product unitary operations U U . Since 1 k d 1 and the equality holds for≤ k = d, A ⊗ B ≤ ≤ − P P in case of pure vector and the corresponding pure state x↓(1 i d) are components of vector x rearranged i ≤ ≤ (projector) ΨAB ΨAB the coefficients ai are the in decreasing order; y↓(1 i d) are defined similarly. only parameters| ih that are| invariant under{ such} opera- i ≤ ≤ Zeros are appended to the vectors λ(ρA), λ(ρB ) in (34), tions, they completely determine entanglement of bipar- in order to make their dimension equal to the one of λ(ρ). tite pure state. The above inequalities constitute necessary condition In particular, it is very easy to see that pure state for separability of bipartite states in arbitrary dimen- (projector) ΨAB ΨAB is separable iff the vector ΨAB sions in terms of the local and the global spectra of is product.| Equivalentlyih | the rank of either of reduced a state. This criterion is stronger than entropic cri- density matrices ̺A, ̺B is equal to 1, or there is sin- terion (30) and it again supports the view that “sep- gle nonzero Schmidt coefficient. Thus for bipartite pure arable states are more disordered globally than locally” states it is elementary to decide whether the state is sep- (Nielsen and Kempe, 2001). An alternative proof of this arable or not by diagonalizing its reduced density matrix. result have been found (Gurvits and Barnum, 2005). So far we have considered entanglement of pure states. Due to decoherence phenomenon, in laboratories we un- avoidably deal with mixed states rather than pure ones. VI. BIPARTITE ENTANGLEMENT However mixed state still can contain some “noisy” en- tanglement. In accordance with general definition for A. Definition and basic properties n-partite state (II) any bipartite state ̺AB defined on Hilbert space AB = A B is separable (see (Werner, The fundamental question in quantum entanglement H H ⊗H theory is which states are entangled and which are not. Only in few cases this question has simple answer. The simplest is the case of pure bipartite states. In accor- 25 i Here the orthonormal basis eX spans subspace X , X = dance with definition of multipartite entangled states A, B. {| i} H (Sec. (II) any bipartite pure state Ψ = 26 T denotes transposition. | ABi ∈ HAB 21
1989a)) if and only if it can neither be represented nor is a density operator (i.e. has nonnegative spectrum), TB approximated by the states of the following form which means automatically that ̺AB is also a quantum state (It also guarantees positivity of ̺TA defined in k AB analogous way). The operation T , called partial trans- ̺ = p ̺i ̺i , (38) B AB i A ⊗ B pose27, corresponds just to transposition of indices corre- i=1 X sponding to the second subsystem and has interpretation i i as a partial time reversal (Sanpera et al., 1998). If the where ̺ , ̺ are defined on local Hilbert spaces A, A B H state is represented in a block form B. In case of finite dimensional systems, i.e. when H i i dim AB < the states ̺A, ̺B can be chosen to be H ∞ ̺00 ̺01 ... ̺0 dA 1 pure. Then, from the Caratheodory theorem it follows − (see (Horodecki, 1997; Vedral and Plenio, 1998)) that the ̺10 ̺11 ... ̺1 dA 1 ̺AB = − (40) number k in the convex combination can be bounded ...... ̺dA 1 0 ̺dA 1 1 ... ̺dA 1 dA 1 by the square of the dimension of the global Hilbert − − − − 2 2 space: k dAB = (dAdB) where dAB = dim AB ≤ H with block entries ̺ i I ̺ j I, then one has etc. It happens that for two-qubits the number of states ij ≡ h |⊗ | AB| i⊗ (called sometimes cardinality) needed in the separable T T T decomposition is always four which corresponds to di- ̺00 ̺01 ... ̺0 dA 1 mension of the Hilbert space itself (see (Sanpera et al., ̺T ̺T ... ̺T − ̺Γ = 10 11 1 dA 1 . (41) 1998; Wootters, 1998)). There are however d d states AB ...... − ⊗ that for d 3 have cardinality of order of d4/2 (see ̺T ̺T ... ̺T ≥ dA 1 0 dA 1 1 dA 1 dA 1 (DiVincenzo et al., 2000b)). − − − − We shall restrict subsequent analysis to the case of Thus the PPT condition corresponds to transposing finite dimensions unless stated otherwise. block elements of matrix corresponding to second sub- The set AB of all separable states defined in this way system. PPT condition is known to be stronger than S is convex, compact and invariant under the product uni- all entropic criteria based on Renyi α-entropy (V) for tary operations UA UB. Moreover the separability prop- α [0, ] (Vollbrecht and Wolf, 2002b). A fundamental ⊗ erty is preserved under so called (stochastic) separable fact∈ is (Horodecki∞ et al., 1996a; Peres, 1996b) that PPT operations (see Sec. XI.B). condition is necessary and sufficient condition for sepa- The problem is that given any state ̺AB it is very hard rability of 2 2 and 2 3 cases. Thus it gives a complete to check whether it is separable or not. In particular, characterization⊗ of separability⊗ in those cases (for more its separable decomposition may have nothing common details or further improvements see Sec. VI.B.2). with the eigendecomposition, i.e. there are many sepa- rable states that have their eigenvectors entangled, i.e. nonproduct. 2. Separability via positive, but not completely positive maps It is important to repeat, what the term entanglement means on the level of mixed states: all states that do not belong to , i.e. are not separable (in terms of the above Peres PPT condition initiated a general analysis of definition)S are called entangled. the problem of the characterization of separable (equiva- In general the problem of separability of mixed states lently entangled) states in terms of linear positive maps appears to be extremely complex, as we will see in the (Horodecki et al., 1996a). Namely, it can be seen that the PPT condition is equivalent to demanding the pos- next section. The operational criteria are known only in 28 special cases. itivity of the operator [IA TB](̺AB), where TB is the transposition map acting on⊗ the second subsystem. The transposition map is a positive map (i.e. it maps B. Main separability/entanglement criteria in bipartite case any positive operator on B into a positive one), but it is not completely positiveH29. In fact, I T is not a A ⊗ B 1. Positive partial transpose (PPT) criterion positive map and this is the source of success of Peres criterion. Let us consider the characterization of the set of mixed It has been recognized that any positive (P) but not completely positive (CP) map Λ : ( ) ( ′ ) with bipartite separable states. Some necessary separability B HB →B HA conditions have been provided in terms of entropic in- equalities, but a much stronger criterion has been pro- vided by Asher Peres (Peres, 1996b), which is called pos- 27 TB Γ itive partial transpose (PPT) criterion. It says that if Following (Rains, 1998) instead of ̺AB we will write ̺AB (as the ̺ , is separable then the new matrix ̺TB with matrix symbol Γ is a right “part” of the letter T ). AB AB 28 The operator is called positive iff it is Hermitian and has non- elements defined in some fixed product basis as: negative spectrum. 29 The map Θ is completely positive iff I Θ is positive for identity T m µ ̺ B n ν m ν ̺ n µ (39) map I on any finite-dimensional system.⊗ h |h | AB | i| i ≡ h |h | AB| i| i 22
W codomain related to some new Hilbert space A′ provides nontrivial necessary separability criterion inH the form: [I Λ ](̺ ) 0. (42) A ⊗ B AB ≥ ̺ent This corresponds to nonnegativity of spectrum of the fol- Tr(W ̺ent) < 0 lowing matrix: ̺sep [I Λ ](̺ ) A ⊗ B AB Λ(̺00) ... Λ(̺0 dA 1) − Λ(̺10) ... Λ(̺1 dA 1)) = − (43) Tr(W ̺sep) ≥ 0 ...... Λ(̺ ) ... Λ(̺ ) dA 1 0 dA 1 dA 1 − − − FIG. 2 The line represents hyperplane corresponding to the with ̺ij defined again as in (40). entanglement witness W . All states located to the left of It happens that using the above technique one can pro- the hyperplane or belonging to it (in particular all separable vide a necessary and sufficient condition for separability states) provide nonnegative mean value of the witness, i.e. Tr(W̺sep) ≥ 0 while those located to the right are entangled (see (Horodecki et al., 1996a)): the state ̺AB is separa- ble if and only if the condition (42) is satisfied for all P states detected by the witness. but not CP maps Λ : ( ) ( ) where , B HB → B HA HA HB describe the left and right subsystems of the system AB. (i) Note that the set of maps can be further restricted where ΛCP stand for some CP maps and T stands for to all P but not CP maps that are identity-preserving transposition. It can be easily shown (Horodecki et al., (Horodecki, 2001b) (the set of witnesses can be then also 1996a) that among all decomposable maps the transpo- restricted via the isomorphism). One could also restrict sition map T is the “strongest” map i.e. there is no de- the maps to trace preserving ones, but then one has to composable map that can reveal entanglement which is enlarge the codomain (Horodecki et al., 2006d). not detected by transposition. Given characterization in terms of maps and witnesses it was natural to ask about a more practical characteriza- tion of separability/entanglement. The problem is that in 3. Separability via entanglement witnesses general the set of P but not CP maps is not characterized and it involves a hard problem in contemporary linear Entanglement witnesses (Horodecki et al., 1996a; algebra (for progress in this direction see (Kossakowski, Terhal, 2000a) are fundamental tool in quantum entan- 2003) and references therein). glement theory. They are observables that completely However for very low dimensional systems there is sur- characterize separable states and allow to detect en- prisingly useful solution (Horodecki et al., 1996a): the tanglement physically. Their origin stems from geome- try: the convex sets can be described by hyperplanes. states of dA dB with dAdB 6 (two-qubits or qubit- qutrit systems)⊗ are separable≤ if and only if they are This translates into the statement (see (Horodecki et al., PPT. Recently even a simpler condition for two-qubit 1996a; Terhal, 2000a)) that the state ̺AB belongs to the systems (and only for them) has been pointed out set of separable if it has nonnegative mean value (Augusiak et al., 2006) which is important in context of Tr(W̺ ) 0 (46) physical detections (see Secs. VIII.C.2 and VIII.C.3) : AB ≥ two-qubit state ̺AB is separable iff for all observables W that (i) have at least one negative Γ eigenvalue and (ii) have nonnegative mean value on prod- det(̺AB) 0. (44) ≥ uct states or — equivalently — satisfy the nonnegativity This is the simplest two-qubit separability condition. It condition is a direct consequence of two facts known earlier but never exploited in such a way: the partial transpose ψA φB W ψA φB 0. (47) of any entangled two-qubit state (i) is of full rank and h |h | | i| i≥ has only one negative eigenvalue. (Sanpera et al., 1998; for all pure product states ψA φB The observables Verstraete et al., 2001a). Note that some generalizations W satisfying conditions (i) and| (ii)i| abovei 30 have been of (44) for other maps and dimensions are also possible named entanglement witnesses and their physical impor- (Augusiak et al., 2006). tance as entanglement detectors was stressed in (Terhal, The sufficiency of the PPT condition for separability 2000a) in particular one says that entanglement of ̺ is in low dimensions follows from the fact (Størmer, 1963; detected by witness W iff Tr(W̺) < 0, see fig. 2. (We Woronowicz, 1976) that all positive maps Λ : ( d) ′ d B C → ( ) where d = 2, d′ = 2 and d = 2, d′ = 3 are decom- posableB C , i.e. are of the form: 30 The witnesses can be shown to be isomorphic to P but not CP (1) (2) Λdec =Λ +Λ T, (45) maps, see Eq. (49). CP CP ◦ 23
shall discuss physical aspects of entanglement detection states this set is also convex, compact and invariant under in more detail subsequently). An example of entangle- product unitary operations. It has been also found that ment witness for d d case is (cf. (Werner, 1989a)) the stochastic separable operations preserve PPT property ⊗ Hermitian swap operator (Horodecki et al., 1998a). In general we have ( P P T . As described in previous section the two setsS areS equal d 1 − for dAdB 6. In all other cases, they differ (Horodecki, V = i j j i . (48) ≤ | ih | ⊗ | ih | 1997) (see Sec. VI.B.7 for examples), i.e. there are en- i,j=0 X tangled states that are PPT. The latter states give rise to the so-called bound entanglement phenomenon (see Sec. To see that V is an entanglement witness let us note XII). that we have ψ φ V ψ φ = ψ φ 2 0 which A B A B A B To describe it is enough to consider only a sub- ensures propertyh |h (ii)| above.| i| i At the|h same| i| time≥ V = P P T set of decomposableS witnesses where P = 0 and Q is P (+) P ( ) where P (+) = 1 (I + V ) and P ( ) = 1 (I V ) − 2 − 2 a pure projector corresponding to entangled vector Φ . correspond− to projectors onto symmetric and antisym-− This gives a minimal set of entanglement witnesses that| i metric subspace of the Hilbert space d d respectively. describe set of PPT states. The required witnesses are Hence V also satisfies (i) since it hasC ⊗C some eigenvalues thus of the form equal to 1. It is interesting that V is an example of so − called decomposable entanglement witness (see (45) and W = Φ Φ Γ (53) analysis below). | ih | The P but not CP maps and entanglement witnesses with some entangled vector Φ . The swap V is propor- are linked by so called Choi-Jamio lkowski isomorphism | i Γ tional to a witness of this kind. Indeed, we have V = dP+ (Choi, 1982; Jamio lkowski, 1972a): (hence — as announced before — the swap is a decom- posable witness). W = [I Λ](P +) (49) Λ ⊗ d For d d systems there is one distinguished decom- posable witness⊗ which is not of the form (53) but is very with pure projector useful and looks very simple. This is the operator + + + Pd = Φd Φd , (50) 1 | ih | W (P +)= I P +. (54) + d − where state vector Φd A A is defined as ∈ H ⊗ H + One can prove that the condition W (P ) ̺PPT 0 pro- d 1 h i ≥ 1 − vides immediately the restriction on the parameter called Φ+ = i i , d = dim . (51) 32 | d i √ | i ⊗ | i HA fidelity or singlet fraction : d i=0 X F (̺) = Tr[P +̺]. (55) + The pure projector Pd is an example of maximally en- 31 tangled state on the space A A. Namely (see (Rains, 2000)): The very important observationH ⊗ isH that while the con- dition (46) as a whole is equivalent to (42), a particular 1 F (̺ ) . (56) witness is not equivalent to a positive map associated via P P T ≤ d isomorphism: the map proves a stronger condition (see discussion further in the text). In particular this inequality was found first for separa- As we already said, the special class of decomposable ble states and its violation was shown to be sufficient P but not CP maps (i.e. of the form 45) which provide for entanglement distillation (Horodecki and Horodecki, no stronger criterion than the PPT one, is distinguished. 1999). Consequently, all the corresponding entanglement wit- As we have already mentioned, the set of maps con- nesses are called decomposable and are of the form (see ditions (42) is equivalent to set of witnesses condition (Lewenstein et al., 2000)): (46). Nevertheless any single witness WΛ condition is much weaker than the condition given by the map Λ. W dec = P + QΓ, (52) This is because the first is of scalar type, while the sec- ond represents an operator inequality condition. To see where P , Q are some positive operators. It can be easily this difference it is enough to consider the two-qubit case shown that decomposable witnesses (equivalently — de- and compare the transposition map T (which detects all composable maps) describe the new set P P T of all states entanglement in sense of PPT test) with the entangle- that satisfy PPT criterion. Like the setS of separable ment witness isomorphic to it, which is the swap op- S eration V , that does not detect entanglement of any
31 For simplicity we will drop the dimension denoting projector onto Φ+ Φ+ as P + P + provided it does not lead to ambiguity. 32 One has 0 F (̺) 1 and F (̺)=1 iff ̺ = P +. ≡ d ≡ d ≤ ≤ 24 symmetric pure state. Indeed it is not very difficult a theory similar to that for “simple” entanglement, with to see (see (Horodecki and Ekert, 2002)) that the con- Schmidt-number witnesses in place of usual witnesses. dition based on one map Λ is equivalent to a continu- The family of sets satisfies inclusion relations: Sk S1 ⊂ ous set of conditions defined by all witnesses of the form 2 ... rmax . Note that here 1 corresponds just to d S ⊂ ⊂ S S WΛ,A A IWΛA† I where A are operators on of the set of separable states, while rmax — to the set of all rank more≡ ⊗ than one.⊗ Thus after a bit of analysisC it is states. Each set is compact, convexS and again closed un- not difficult to see why PPT condition associated with a der separable operations. Moreover, each such set is de- single map (transposition T) is equivalent to set of all the scribed by k-positive maps (Terhal and Horodecki, 2000) conditions provided by the witnesses of the form (53). or by Schmidt-rank k witnesses (Sanpera et al., 2001). A On the other hand one must stress condition based on Schmidt rank k witness is an observable Wk that satis- a witness is naturally directly measurable (Terhal, 2000b) fies two conditions (i) must have at least one negative (and it has found recently many experimental implemen- eigenvalue and (ii) must satisfy: tations, see introduction section) while physical imple- mentation of separability condition based on (unphysical) Ψk Wk Ψk 0, (57) h | | i≥ P but not CP maps is much more complicated, though still possible (see Sec. VIII.C.1). Moreover, the power of for all Schmidt rank k vectors Ψk AB. As in case ∈ H witnesses can be enhanced with help of nonlinear correc- of separability problem the k-positive maps are related tions (see Sec. VIII.B). via Choi-Jamio lkowski isomorphism to special maps that For higher dimensional systems there are many nonde- are called k-positive (i.e. such that [Ik Λk] is posi- tive for I being identity on ( k)) but not⊗ completely composable P but not CP maps but their construction k B C is in general hard (see (Kossakowski, 2003) and reference positive. The isomorphism is virtually the same as therein). In particular checking that given map is P is the one that links entanglement witnesses W = W1 very hard and equivalent to checking the positivity con- with k-positive maps Λ1. Important maps (respectively dition (ii) in the definition of witnesses. Note that long witnesses) are the ones which discriminate between k and . They are those maps, which are k-positiveS time ago in literature an algorithm that corresponds to Sk+1 checking strict positivity of witnesses was found (ie. (ii) but not k + 1 positive. A nice example is the fol- condition with strict inequality) (Jamio lkowski, 1972b)) lowing family of maps Λp(̺) = ITr(̺) p(̺) which is k but not k + 1 positive for 1 <− p 1 (see but its complexity is very high (it has an interesting con- k+1 ≤ k sequences though which will be mentioned later in more (Terhal and Horodecki, 2000)). The case of p = 1 cor- details). responds to special so called reduction map which plays The important question one can ask about en- a role in entanglement distillation defining in particular tanglement witnesses regards their optimality reduction separability criterion which will be discussed (Lewenstein et al., 2000, 2001). We say that an subsequently. Many techniques have been generalized entanglement witness W1 is finer than W2 iff the entan- from separability to mixed state Schmidt rank analysis glement of any ̺ detected by W2 is also detected by W1. (see (Sanpera et al., 2001)). For general review of sep- A given witness W is called optimal iff there is no witness arability problem including especially entanglement wit- finer than it. The useful sufficient condition of optimality nesses see (Bruß, 2002; Bruß et al., 2002; Terhal, 2002). (Lewenstein et al., 2000) is expressed in terms of the The Schmidt number witnesses and maps description is Hilbert subspace = φ ψ : φ ψ W φ ψ = 0 . reviewed extensively in (Bruß et al., 2002). PW {| i| i h |h | | i| i } Namely if W spans the whole Hilbert space then the witness is optimal.P In a sense it is then fully “tangent” to set of separable states. The systematic method of 4. Witnesses and experimental detection of entanglement optimization of given entanglement witness have been worked out first in (Lewenstein et al., 2000, 2001) (for As already mentioned, entanglement witnesses have alternative optimization procedure see (Eisert et al., been found very important in experimental detection of 2004), cf. optimization of witnesses for continuous entanglement (Horodecki et al., 1996a; Terhal, 2000b): variables (Hyllus and Eisert, 2006)). for any entangled state ̺ent there are witnesses which In some analogy to the pure bipartite case, are signatures of entanglement in a sense that they are we can define the Schmidt rank for density ma- negative on this state Tr(W̺ent) < 0. Here we shall de- trices (Terhal and Horodecki, 2000) as rS (̺) = scribe few further aspects of the detection showing both min(maxi[rS(ψi)]) where minimum is taken over the importance of entanglement witnesses and their pos- all decompositions ̺ = p ψ ψ and r (ψ ) are the sible applications. i i| iih i| S i Schmidt ranks of the corresponding pure states (see Sec. The first issue concerns the following question VI.A). One can easilyP prove that separable operations (Horodecki et al., 1999c): Given experimental mean val- can not increase it. ues of incomplete set of observables Ai = ai what infor- Now, for any k in the range 1,...,r with r = mation about entanglement shouldh bei concluded basing { max} max min[dA, dB], we have a set k of states with Schmidt on those data? The idea of the paper was that if entan- number not greater than k.S For such sets we can build glement is finally needed as a resource then the observer 25
33 should consider the worst case scenario, i.e. should min- count partial separability as well). Now if WO has imize entanglement under experimental constraints. In negative eigenvalue becomes immediately an entangle- other words, experimental entanglement should be of the ment witness by construction. In case of spin lattices form: one takes O = H where H is a Hamiltonian of the system and calculates W for quantum Gibbs state h H i̺ k ̺Gibbs = exp( H/kT )/Tr[exp( H/kT )]. It can be im- E(a1,...,ak) = inf Ai ̺=ai E(̺). (58) {h i }i=1 mediately seen− that for H with− discrete spectrum the ob- Such minimization of entanglement of formation and rel- servable WH has a negative eigenvalue iff the lowest en- ative entropy of entanglement was performed for given ergy state is entangled and then the observable becomes mean of Bell observable on unknown 2-qubit state. It is entanglement witness by construction (see (59)). In this interesting that in general there are many states achiev- one can estimate the range of temperatures for which the mean value W is negative (Toth, 2005). Fur- ing that minimum — to get single final state the authors h H i̺Gibbs also have proposed final application of maximum entropy ther improvements involve uncertainty based entangle- Jaynes principle (Horodecki et al., 1999c). ment witnesses (Anders et al., 2005) and applications of entanglement measures like robustness of entanglement Quite recently the idea of minimization of entan- (Markham et al., 2006) to thermal entanglement. glement under experimental constraints was applied with help of entanglement witnesses (Eisert et al., 2007; Finally let us recall an important issue: how to decom- G¨uhne et al., 2006b). In (G¨uhne et al., 2006b) using con- pose given witness into locally measurable observables vex analysis the authors have performed minimization of (G¨uhne et al., 2002), (G¨uhne et al., 2003). This is an convex entanglement measures for given mean values of important issue since if we want to detect entanglement entanglement witnesses basing on approximation of con- between spatially separated systems we can only measure vex function by affine functions from below. Specific es- mean values of tensor products of local observables con- timates have been performed for existing experimental sistent with spatial separation (cf. Sec. XI). For a given data. Independently a similar analysis of lower bounds witness W in dA dB one can ask about the minimal for many entanglement measures has been performed in number of local measurements⊗ on systems A, B that can (Eisert et al., 2007), where the emphasis was put on an- reconstruct the mean value of the whole witness. The alytical formulas for specific examples. The derived for- problem is to find optimal decomposition mulas (Eisert et al., 2007; G¨uhne et al., 2006b) provide a direct quantitative role for results of entanglement wit- nesses measurements. Note that more refined analysis was focused on correlations obtained in the experiment, r W = γ Xk Y k (60) identifying which types of correlations measured in in- k A ⊗ B complete experiments may be already signature of en- kX=1 tanglement (Audenaert and Plenio, 2006). Another experimental issue, where entanglement wit- nesses have been applied, is the problem of macroscopic onto a set of product of normalized (in Hilbert-Schmidt entanglement at finite temperature. A threshold tem- norm) observables such that any two pairs Xk Y k, ′ ′ A B perature for existence of entanglement can be identi- k k ⊗ XA YB in the sum differ on at most one side. Then fied. The relation between thermal equilibrium state the cardinality⊗ of the representation (60) is calculated as and entanglement was hidden already in 2-qubit analysis s = ′r rI′ where rI is a number of those product terms of Jaynes principle and entanglement (Horodecki et al., k − k rI X Y ′ in which at least one of local observables A B k =1 ′ 1999c). The first explicit analysis of entanglement in { ⊗ } k is proportional to identity I (say XA = αI) and the sec- thermal state was provided by Nielsen (Nielsen, 1998) ′ ond one (Y k ) is linearly dependent on all the other local where first calculation of temperatures for which entan- B ′ k k glement is present in two-qubit Gibbs state was per- ones (i.e. YB = k=k′ αkYB ). The optimal cardinality 6 formed. A fundamental observation is that entanglement smin minimize over all decompositions of the type (60) P witnesses theory can be exploited to detect entanglement gives minimal number of different measurement settings in general (multipartite) thermal states including sys- needed to measure mean of given entanglement between tems with large number of particles (Brukner and Vedral, spatially separated systems. The problem of finding the 2004; Toth, 2005). In the most elegant approach, for any optimal decomposition (60) with the minimal cardinality observable O one defines entanglement witness as follows smin has been investigated in papers (G¨uhne et al., 2002, (see (Toth, 2005)): 2003) and analytical solutions have been found for both bipartite cases discussed above as well as for multipartite generalizations. WO = O inf Ψprod O Ψprod , (59) − Ψprod h | | i | i where infimum is taken over all product pure states Ψprod (note that the method can be extended to take into ac- 33 See Sec. VII. 26
5. Entanglement witnesses and Bell inequalities the so called reduction criterion (Cerf et al., 1999; Horodecki and Horodecki, 1999) defined by the formula Entanglement witnesses (see Sec. VI) are Hermitian (42) with the reduction map:Λred(̺) = ITr(̺) ̺. operators that are designed directly for detection of en- This map is decomposable but — as we shall see− sub- tanglement. In 2000 Terhal first considered a possible sequently — plays important role in entanglement dis- connection between entanglement witnesses and Bell in- tillation theory(Horodecki and Horodecki, 1999). Only equalities (Terhal, 2000a). From a “quantum” point of in case of two-dimensional Hilbert space the map rep- view, Bell inequalities are just nonoptimal entanglement resents just a reflection in Bloch sphere representa- witnesses. For example one can define the CHSH-type tion (Bengtsson and Zyczkowski,˙ 2006) and can be eas- witness which is positive on all states which admit LHVM ily shown to be just equal to transpose map T followed by σy i.e. σyT (̺)σy. As such it provides a separability WCHSH =2 I CHSH , (61) condition completely equivalent to PPT in this special · −B (two-qubit) case. In general the reduction separability where CHSH is the CHSH operator (16). Such defined red red B criterion [IA ΛB ](̺AB ) 0 generated by Λ can be witness is of course nonoptimal one since the latter is written as: ⊗ ≥ strictly positive on separable states. However, basing on the concept of the optimal witness (Lewenstein et al., ̺A I ̺AB 0 (62) 2000) one can estimate how much optimal witnesses ⊗ − ≥ have to be shifted by the identity operator to make and since Λred is decomposable them positive on all states admitting a LHVM. More- (Horodecki and Horodecki, 1999) the corresponding over, there exists a natural decomposition of the CHSH separability criterion is weaker than PPT one (see Sec. witness into two optimal witnesses, and the identity op- VI.B.2). On the other hand it is interesting that this erator (Hyllus et al., 2005). In multipartite case, Bell in- criterion is stronger (Hiroshima, 2003) than mixing equalities can even detect so called bound entanglement separability criteria (Nielsen and Kempe, 2001) as well (Augusiak et al., 2006; D¨ur, 2001; Kaszlikowski et al., as some entropic criteria with α [0, 1] and α = 2000; Sen(De) et al., 2002) (see Sec. XII.J). (for the proofs see (Vollbrecht and∈ Wolf, 2002b) and∞ Inspired by CHSH inequalities Uffink proposed in- (Horodecki and Horodecki, 1999) respectively). equalities (Uffink, 2002), which are no longer im- Another very important criterion is the one based plied by LHVM, yet constitute a (nonlinear) entangle- on the map due to Breuer and, independently, Hall ment witness (see Sec. VIII.B), see in this context (Breuer, 2006a; Hall, 2006) which is a modification of (Uffink and Seevinck, 2006). reduction map on even dimensional Hilbert space d = Svetlichny and Sevnick proposed Bell inequalities that 2k. On this subspace there exist antisymmetric uni- can be used as a detector of genuinely multipartite en- tary operations U T = U (for instance the one U = tanglement (Seevinck and Svetlichny, 2002). Toth and antidiag[1, 1, 1, 1,...,−1, 1] (Breuer, 2006a)). The G¨uhne proposed a new approach to entanglement detec- corresponding− antiunitary− map− U( )T U, maps any pure tion (Toth and G¨uhne, 2005a,b)based on the stabilizer state to some state that is orthogonal· to it. This leads theory (Gottesman, 1996). In particular, they found to the conclusion that the map which acts on the state ̺ interesting connections between entanglement witnesses as follows: and Mermin-type inequalities (Toth and G¨uhne, 2005b). red T In general, the problem of the relation between Bell in- Λ(̺)=Λ (̺) U(̺) U † (63) equalities and entanglement witnesses is very complex. It − follows from the very large number of degrees of freedom is positive for any antisymmetric U. This map is not of the Bell inequalities. Nevertheless it is basic problem, decomposable and the entanglement witness WΛ corre- as the Bell observable is a double witness. It detects not sponding to it has an optimality property since the corre- only entanglement but also nonlocality. sponding space WΛ (see Sec. VI.B.3) is the full Hilbert P It is interesting that the loophole problem in the ex- space (Breuer, 2006b). This nondecomposability prop- perimental tests of Bell inequalities (see (Gill, 2003)) has erty allows the map to detect special class of very weak found its analogy (Skwara et al., 2006) in an entangle- entanglement, namely PPT entanglement mentioned al- ment detection domain. In particular the efficiency of ready before. We shall pass to its more detailed descrip- detectors that still allows to claim that entanglement was tion now. detected, has been analyzed and related to cryptographic application. 7. Range criterion and its applications; PPT entanglement
6. Distinguished maps criteria: reduction criterion and its The existence of nondecomposable maps (witnesses) extensions for the cases with dA dB > 6 implies that there are states that are entangled· but PPT in all those cases. Thus There are two important separability criteria pro- the PPT test is no longer a sufficient test of separabil- vided by P but not CP maps. The first one is ity in those cases. This has striking consequences, for 27 quantum communication theory, including entanglement leads to construction of PPT entangled states. This re- distillation and quantum key distribution which we dis- sult was further exploited to provide new nondecompos- cuss further in this paper. Existence of PPT entangled able maps (Terhal, 2000b). What was the mathematical states was known already in terms of cones in mathemat- origin of the construction? This is directly related to the ical literature (see for example (Choi, 1982)) sometimes question: what is the other way PPT entanglement can expressed in direct sum language. be detected? As we already announced there is a rule On physical ground, first examples of entangled states linked directly to the positive-map separability condition that are PPT were provided in (Horodecki, 1997), follow- (see (Horodecki et al., 1996a)): for any PPT entangled ing Woronowicz construction (Woronowicz, 1976). Their state there is a nondecomposable P but not CP map Λ entanglement had to be found by a criterion that is in- such that the criterion (42) is violated. Thus a way to dependent on PPT one. As we already mentioned, this detect PPT entanglement is to find a proper nondecom- might be done with properly chosen P but not CP non- posable P but not CP map. The alternative statement, decomposable map (see Sec. VI.B.2). In (Horodecki, saying that PPT state is entangled if an only if it is de- 1997) another criterion was formulated for this purpose, tected by some nondecomposable witness (i.e. the one which is useful for other applications (see below). This is that is not of the form (52)) is immediately also true. the range criterion: if ̺AB is separable, then there exists It is rather hard to construct nondecomposable maps a set of product vectors ψi φi , such that it spans and witnesses respectively. En example of P but nor CP { A ⊗ B} i i TB nondecomposable map due to Choi is (Choi, 1982) (see range of ̺AB while ψA (φB )∗ spans range of ̺AB, where complex conjugate{ ⊗ is taken} in the same basis in (Kossakowski, 2003) and references therein for general- which PPT operation on ̺AB has been performed. In ization): particular an example of 3 3 PPT entangled state re- ⊗ a a a a + a a a vealed by range criterion (written in a standard basis) 11 12 13 11 33 − 12 − 13 was provided: Λ( a21 a22 a23 )= a21 a22 + a11 a23 , a a a −a a a −+ a 31 32 33 − 31 − 32 33 22 a 0 0 0 a 0 0 0 a (65) 0 a 0000 0 0 0 0 0 a 000 0 0 0 which allows to detect PPT entangled states that help 0 0 0 a 00 0 0 0 in bound entanglement activation (see (Horodecki et al., 1 ̺ = a 0 0 0 a 0 0 0 a , (64) 1999a)). a 8a +1 00000 a 0 0 0 The new technique for achieving the nondecompos- 1+a √1 a2 able maps was worked out by Terhal (Terhal, 2000b). 000000 0 − 2 2 000000 0 a 0 Her idea was to take a projector on UPB space 2 √1 a 1+a P UP B and observe that the following quantity ǫ = a 0 0 0 a 0 2− 0 2 H minΨsep Ψsep P UP B Ψsep is strictly positive because of h | H | i where 0 “nonvanishing” admixture of edge state. An example of the first proposal of systematic way of checking separa- edge state is just any state based on UPB construction bility. ̺UP B . Another edge state is 2 4 PPT entangled state Similar technique has been used to find the decompo- ⊗ from (Horodecki, 1997). sition of PPT entangled states. Namely it happens that Now generalization of Terhal construction leads to a any PPT entangled state can be decomposed into the method that in fact detects any PPT entanglement is form ̺ = (1 p)̺sep + pδedge where δedge is an edge state (Lewenstein et al., 2001): defined above.− This is a systematic method leading to ǫ necessary and sufficient separability test for states which W = P + QΓ C, (67) are PPT i.e. in the region where checking separability − c is the hardest (the state is separable if at the end the with P , Q being positive operators supported on ker- parameter p is zero). Same difficulty like in BSA method Γ nels of δedge and δedge respectively while the parame- is finding product vector in the range of the matrix. The Γ ter ǫ = minΨsep Ψsep P + Q Ψsep can be shown to be problem becomes much more tractable in case of states strictly positiveh (by extremal| | violationi of range criterion that satisfy low rank condition (Horodecki et al., 2000e) Γ by δ state) while C is arbitrary positive operator with r(̺)+ r(̺ ) 2dAdB dA dB + 2. Then typically (in edge ≤ − − Tr(δedge) > 0 and c = maxΨsep Ψsep C Ψsep . All the so called generic case) the state has only finite number above witnesses are nondecomposableh |and| it isi interest- of product states satisfying the range criterion which can ing that entanglement of all PPT entangled states can be be found by solving polynomial equations. In such cases detected even by a restricted subclass of the above, when the separability problem can be solved in finite number P , Q are just projectors on the kernels of δedge while C of steps (Horodecki et al., 2000e). is the identity operator (then c = 1). Of course all maps Let us also mention that recently in analogy to the isomorphic to the above witnesses are also nondecompos- BSA construction above the decomposition onto the sym- able. metrically extendible and nonextendible parts has been We would like also to mention a nice idea of con- found which has a very nice cryptographic application struction of nondecomposable map based on k-positive (Moroder et al., 2006b) (see Sec. XIX). The BSA con- maps. Namely it happens that for any map Λk that struction leads also to an entanglement measure (see Sec. is k-positive but not k + 1 positive (cf. application in XV.C.4). Schmidt rank of density matrix (Terhal and Horodecki, 2000)) the positive (by definition) map I Λ is al- k ⊗ k ready nondecomposable (Piani and Mora, 2006). This is 8. Matrix realignment criterion and linear contractions criteria also connected to another construction of PPT entan- gled states (Piani and Mora, 2006) which was inspired There is yet another strong class of criteria based by very useful examples due to (Ishizaka, 2004) in pure on linear contractions on product states. They stem states interconvertibility. from the new criterion discovered in (Rudolph, 2003), Coming back to range criterion introduced here, (Chen and Wu, 2003) called computable cross norm there is an interesting application: the so called (CCN) criterion or matrix realignment criterion which is Lewenstein-Sanpera decomposition. Namely any bi- operational and independent on PPT test (Peres, 1996b). partite state ̺ can be uniquely decomposed (see In terms of matrix elements it can be stated as follows: (Karnas and Lewenstein, 2000)) in the following way if the state ̺AB is separable then the matrix (̺) with (Lewenstein and Sanpera, 1998): elements R ̺ = (1 p)̺sep + pσ, (68) − m µ (̺AB ) n ν m n ̺ ν µ (69) h |h |R | i| i ≡ h |h | | i| i where ̺sep (called best separable approximation BSA) is a separable state, σ is entangled and p∗ is a max- has trace norm not greater than one (there are many imal probability p∗ [0, 1] such that the decomposi- other variants see (Horodecki et al., 2006d). tion of the above form∈ but with σ taken to be arbi- It can be formally generalized as follows: if Λ satisfies trary state is still true. Clearly ̺ is separable iff p∗ = 1. For two-qubits the entangled part σ is always pure Λ( φA φA φB φB ) 1 1 (70) k | ih | ⊗ | ih | k ≤ (Lewenstein and Sanpera, 1998). Moreover the decom- position can be then found in a fully algebraic way with- for all pure product states φA φA φB φB then for | ih | ⊗ | ih | 34 out optimization procedure (Wellens and Ku´s, 2001). In any separable state ̺AB one has Λ(̺AB) 1 1 . The k k ≤ particular if ̺ is of full rank, then σ is maximally en- matrix realignment map which permutes matrix el- sep R tangled and p∗ is just equal to so called Wootters con- ements just satisfies the above contraction on products currence (see (Wellens and Ku´s, 2001) for the proof). In criterion (70). To find another interesting contractions general the way of construction of the decomposition re- quires technique of subtracting of product states from the range of the matrix (Lewenstein and Sanpera, 1998) (cf. 34 (Horodecki et al., 2000e; Kraus et al., 2000)). This was Here X 1 = Tr√XX denotes trace norm. k k † 29 of that type that are not equivalent to realignment is an 10. Characterization of bipartite separability in terms of open problem. biconcurrence Quite remarkably the realignment criterion has been found to detect some of PPT entanglement In this section we shall describe a quadratic function (Chen and Wu, 2003)(see also (Rudolph, 2003)) and to of the state that provides necessary and sufficient con- be useful for construction of some nondecomposable dition for separability called biconcurrence. This func- maps. It also provides nice lower bound on concurrence tion was inspired by a generalization of two-qubit Woot- function (see (Chen et al., 2005b)). On the other hand ters’ concurrence due to (Rungta et al., 2001), that ex- it happens that for any state that violates the realign- ploited the universal state inverter, which in turn is ment criterion there is a local uncertainty relation (LUR) actually the reduction map Λred (see Sec. VI.B.6). (see Sec. VIII.A) that is violated but converse statement The generalized concurrence can be written in the form is not always true (G¨uhne et al., 2006a). On the other 1 red red C(ψAB) = ψAB [ΛA ΛB ]( ψAB ψAB ) ψAB = hand finding LUR-s (like finding original entanglement 2 h | ⊗ | ih | | i q2 witnesses) is not easy in general and there is no prac- 2(1 Tr(̺B), which directly reproduces Wootters’ concurrence− in case of two-qubits (see Sec. XV.C.2.b). tical characterization of LUR-s known so far, while the p realignment criterion is elementary, fast in application In (Badzi¸ag et al., 2002), a simplified form was obtained: and still powerful enough to detect PPT entanglement. C(ψ) = ψ [I Λred]( ψ ψ ) ψ h AB | A ⊗ B | AB ih AB | | ABi q 9. Some classes of important quantum states: entanglement = 1 Tr(̺2 ). (73) − B regions of parameters q Now for any ensemble realizing mixed state ̺ = In this section we shall recall classes of states or which k ˜i ˜i 2 i=1 pi ψAB ψAB with k N := (dAdB) and ψi := PPT property is equivalent to separability. i | ih | ≤ | i √pi ψ˜ , the N N biconcurrence matrix Bmµ,nν is P AB We shall start from Werner states that are linked defined| as:i ⊗ to one of the most intriguing problem of entangle- red ment theory namely NPT bound entanglement problem Bmµ,nν ψm [I Λ ]( ψµ ψn ) ψν (74) (DiVincenzo et al., 2000a; D¨ur et al., 2000a) (see Sec. ≡ h | ⊗ | ih | | i XII). (we have dropped here the subsystem indices AB and Werner d d states (Werner, 1989a) .- Define projec- extended the ensemble to N-element one by adding extra (+) ⊗ ( ) 35 tors P = (I+ V )/2, P − = (I V )/2 with identity I N k zero vectors). It can be written equivalently as , and “flip” operation V (48): − − ψm ψµ ψn ψν The following d d state Bmµ,nν = ψm ψn ψµ ψν Tr[(A )†A (A )†A ] ⊗ h | ih | i− (75) ψ 2 (+) 2 ( ) with the matrix of coefficients A defined by the relation W (p)=(1 p) P + p P − , 0 p 1 dA 1 dB 1 ψ − d2 + d d2 d ≤ ≤ ψ = − − A i j . Now, there is an important − i=0 j=0 ij | i| i (71) observation namely the state ̺AB is separable if and only is invariant under any U U operation for any unitary U. if theP scalarP biconcurrence function ⊗ 1 W (p) is separable iff it is PPT which holds for 0 p 2 . Isotropic states (Horodecki and Horodecki,≤ 1999)≤ .- N They are U U ∗ invariant (for any unitary U) d d (̺) := inf [U UBU † U †]mm,mm (76) B U ⊗ ⊗ ⊗ ⊗ m=1 states. They are of the form X 1 F F d2 1 vanishes (Badzi¸ag et al., 2002). Here infimum (equal to ̺F = − I+ − P+, 0 F 1 (72) minimum) is taken over all unitary operations U defined d2 1 d2 1 ≤ ≤ − − on Hilbert space (dim = d2 d2 ) while the matrix H H A B (with P + defined by (50)). An isotropic state is separable B represents operator on . As a matter of fact 1 value of the biconcurrence functionH ⊗ H (76) is just the square iff it is PPT which holds for 0 F d . “Low global rank class” (Horodecki≤ ≤ et al., 2000e) .- of Euclidean norm of concurrence vector introduced in The general class of d d state of all states which (Audenaert et al., 2001c) (see sec. XV.C.2.b). A ⊗ B have global rank not greater than local ones: r(̺AB ) It is interesting that, as one can easily show, (̺) = ≤ N 2 2 B max[r(̺A), r(̺B )]. Here again PPT condition is equiva- inf pi,ψi i=1 pi C(ψi) where infimum is taken over all { } lent to separability. In particular for r(̺AB ) = r(̺A) = N-element ensembles realizing given state ̺. Note that P r(̺B ) the PPT property of ̺AB implies separability (Horodecki et al., 2000e). If r(̺AB ) < max[r(̺A), r(̺B )] (which corresponds to violation of entropic criterion for α = ) then PPT test is violated, because re- 35 A simple expression for biconcurrence was exhibited in ( ) ∞ − duction criterion is stronger than S entropy criterion (Mintert et al., 2004). Bm,µ,n,ν = ψm ψµ PAA′ ∞ ( ) h |h | ⊗ (Horodecki et al., 2003f). P − ′ ψn ψν . BB | i| i 30 putting just square root under sum provides the seminal A1...An = A1 ... Am is analogous to that in H H ⊗ ⊗ Hk i i concurrence value for mixed states bipartite case: ̺AB = pi̺ ... ̺ . The i=1 A1 ⊗ ⊗ Am k Caratheodory bound is kept k dim 2 . Such P ≤ HA1...Am (̺) = inf pi,ψi piC(ψi), (77) defined set of m-separable states is again (i) convex and C { } i=1 X (ii) closed (with respect to trace norm) Moreover sepa- which has an advantage of being an entanglement mono- rability is preserved under m-separable operations which tone and can be bounded analytically (Mintert et al., are immediate generalization of bipartite separable ones 2004)). Still, as in all concurrence cases there is no direct 1 n 1 n algorithm known that finds the minimum efficiently even i Ai Ai ̺A1,...,Am (Ai Ai )† ̺A1,...,Am 1⊗···⊗ n 1⊗···⊗ n for low dimensional systems. → Tr( A A ̺ 1 (A A ) ) P i i i A ,...,Am i i † Interestingly, the biconcurrence matrix allows to for- ⊗···⊗ ⊗···⊗(79) mulate separability problem in terms of a single ”entan- The separabilityP characterization in terms of positive but glement witness”, though acting on a completely different not completely positive maps and witnesses generalizes in Hilbert space (Badziag et al., 2007). Namely for a give a natural way (Horodecki et al., 2001a). There is a con- state ρ one constructs a special witness Wρ such that dition analogous to (42) with I acting on first subsystem the state is separable iff this witness vanishes on some and the map Λ : ( ) ( ). HA1 A2...Am B HA2,...,Am → B HA1 product state. Namely, in the formula (42) we take the maps ΛA2...Am : ( ) ( ) that are positive on product B HA2,...,Am → B HA1 states ie. ΛA2...Am ( φA2 φA2 φAm φAm ) 0 11. Enhancing separability criteria by local filters | ih |⊗···⊗| ih | ≥ (with arbitrary states φAi Ai ) but not completely positive. The corresponding∈ entanglement H witness must For strictly positive density matrices one can have again (i) at least one negative eigenvalue and also strengthen separability criteria by use of the fol- satisfy (ii) lowing result of (Leinaas et al., 2006) (see also (ande J. Dehaene and Moor, 2003)). For any such state φA1 ... φAm W φA1 ... φAm 0. (80) ρ there exist invertible operators A and B such that h | h | | i | i≥ 2 d 1 Maps and witnesses are again related by the isomorphism A− 1 (49) with maximally entangled state P+ on bipartite sys- ρ˜ = A BρA† B† = (I + aiEi Fi), (78) ⊗ ⊗ d d ⊗ tem A1A1. A B i=1 X The above description provides full characterization of where Ei (Fi) are traceless orthonormal hermitian oper- m-separability of m-partite system. An example of maps ators. The operation is called filtering (see sec. XI.B)). positive on product states is simply a product of positive The operators A and B can be found constructively. Due maps. Of course there exist maps that are positive on to invertibility of operators A and B, the (unnormal- product states, but are not of the latter form. (those are ized) stateρ ˜ is entangled if and only if the original state in particular maps (Horodecki et al., 2001a) detecting en- is entangled. Thus the separability problem reduces to tanglement of some semiseparable states constructed in checking states of the above form. They have, in par- (Bennett et al., 1999b), see one of examples below). Mul- ticular, maximally mixed subsystems. Given any sep- tipartite witnesses and related maps were investigated in arability criterion, it often proves useful to apply it to (Jafarizadeh et al., 2006) by means of linear program- the filtered stateρ ˜ rather than to original state (see e.g. ming. (G¨uhne et al., 2007)). Example .- An elementary example of fully separable 3-qubit state is: VII. MULTIPARTITE ENTANGLEMENT — 3 3 ̺ = p 0 0 ⊗ + (1 p) 1 1 ⊗ . (81) SIMILARITIES AND DIFFERENCES | ih | − | ih | Now let us come back to the description of general In multipartite case the qualitative definition of sepa- mixed states separability criteria in multipartite systems. rability and entanglement is much richer then in bipartite Note that in this case there is no simple necessary and case. There is the so-called full separability, which is the sufficient condition for separability like PPT characteri- direct generalization of bipartite separability. Moreover, zation of 2 2 or 2 3 case. Even for three qubits no there are many types of partial separability. Below we such criterion⊗ has been⊗ found so far, in particular check- will briefly discuss the separability criteria in this more ing PPT criteria with respect to all bipartite partitions is complicated situation. not enough at all. However there are many criteria that may be applied. Range criterion immediately general- izes to its many variants since now we require range of A. Notion of full (m-partite) separability ̺A1,...Am to be spanned by φA1 ... φAm while range of TA ...A {| i | i} k1 kl The definition of full multipartite separability (or just the state ̺A1...Am partially transposed with respect to m-separability) of m systems A ...A with Hilbert space subset A ,...,A A ,...,A is clearly required 1 m { k1 kl }⊂{ 1 m} 31 to be spanned by the products of these vectors where all However violation of this condition clearly does not au- with indices k1,...,kl are complex conjugated. Of course tomatically guarantee what can be intuitively considered if the state is separable all such partial transposes must as “truly” m-partite entanglement (to understand it see lead to matrices with nonnegative spectrum, more pre- for instance the 4-qubit state ΨA1A2A3A4 = ΦA1A2 TA ...A | i⊗ cisely all matrices of the type ̺ k1 kl should be states ΦA3A4 where at least one vector ΦA1A2 , ΦA3A4 is en- A1...Am | i themselves. tangled). The realignment criteria are generalized to permu- One says that the state is m-partite entangled iff all bi- tational criteria (Chen and Wu, 2002; Horodecki et al., partite partitions produce mixed reduce density matrices 2006d) which state that if state ̺A1A2...An is sep- (note that both reduced states produced in this way have arable then the matrix [ (̺)] the same nonzero eigenvalues). This means that there Rπ i1 j1,i2j2,...,injn ≡ ̺π(i1j1,i2j2,...,injn) (obtained from the original state does not exist cut, against which the state is product. To via permutation π of matrix indices in product basis) this class belong all those pure states that satisfy general- satisfies π(̺) 1 1. Only some permutations give ized Schmidt decomposition (like the GHZ state above). nontrivialkR criteria,k ≤ that are also different from partial But there are many others, e.g. the mentioned W state. transpose. It is possible to significantly reduce the num- In Sec. XIII we will discuss how one can introduce classi- ber of permutations to the much smaller set of those that fication within the set of m-partite entangled states. One provide independent criteria (Wocjan and Horodecki, can introduce a further classification by means of stochas- 2005). In the case of three particles one has a spe- tic LOCC (SLOCC) (see Sec. XIII), according to which cial case of partial realignment (Chen and Wu, 2003; for 3 qubits there are two classes of truly 3-partite entan- Horodecki et al., 2006d)). Finally, let us recall that the gled states, represented just by the GHZ and W states. contraction criterion (70) generalizes immediately. There are furthermore three classes of pure states which Let us now consider the case of pure states in more de- are partially entangled and partially separable: this is tail. A pure m-partite state is fully separable if and only the state Φ+ 0 (where Φ+ = 1 ( 0 0 + 1 1 ) and | i| i √2 | i| i | i| i if it is a product of pure states describing m elemen- its twins produced by two cyclic permutations of subsys- tary subsystems. To check it, it is enough to compute tems. We see that in the case of those pure states only reduced density matrices of elementary subsystems and 2-qubit entanglement is present and explicitly “partial” check whether they are pure. However, if one asks about separability can be seen. This leads us to the various the possible ways this simple separability condition is vi- notions of partial separability which will be described in olated then the situation becomes more complicated. next section. Here we will present an important family The first problem is that in multipartite case (in of pure entangled states. comparison to bipartite one) only very rarely pure Example: quantum graph states. General states admit the generalized Schmidt decomposition form of graph states has been introduced in min[dA1 ,...,dAm ] i i ΨA1,...,A = ai e˜ e˜ (see | m i i=1 | A1 i⊗···⊗| Am i (Raussendorf et al., 2003) as a generalization of (Peres, 1995; Thaplyial, 1999)). An example of the state cluster states (Briegel and Raussendorf, 2001) that have P m admitting Schmidt decomposition in the d⊗ case is the been shown to be a resource for one-way quantum generalized Greenberger-Horne-Zeilinger state computer (Raussendorf and Briegel, 2001). Universality d 1 of quantum computer based on graph states is one of − (m) 1 m GHZ = ( i ⊗ ), (82) the fundamental application of quantum entanglement | id √ | i d i=0 in a theory of quantum computer (see (Hein et al., X which is a generalization of original GHZ state 2005)). In general, any graph state is a pure m-qubit state G corresponding to a graph G(V, E). The graph (Greenberger et al., 1989) that is a three-qubit vector | i GHZ = 1 ( 0 0 0 + 1 1 1 ). To give an example of is described by the set V of vertices with cardinality | i √2 | i| i| i | i| i| i V = m (corresponding to qubits of G ) and the set E state which does not admit Schmidt decomposition, note of| | edges, i.e. pairs of vertices (corresponding| i to pairs of that the latter implies that if we trace out any subsys- qubits of G )36. Now, the mechanism of creating G is tem, the rest is in fully separable state. One easily finds very simple.| i One takes as the initial state + m| withi that the following state ⊗ + = 1 ( 0 + 1 ). Then according to graph|Gi(V, E), to 1 | i √2 | i | i W = ( 0 0 1 + 0 1 0 + 1 0 0 ) (83) any of pairs of qubits corresponding to vertices connected | i √3 | i| i| i | i| i| i | i| i| i by an edge from E one applies a controlled phase gate: has entangled two qubit subsystem, hence does not admit UC phase = 0 0 I+ 1 1 σ3. Note that since all − | ih |⊗ | ih |⊗ Schmidt decomposition (D¨ur et al., 2000b). such operations commute even if performed according to Thus, in general entanglement of pure state is de- scribed by spectra of the reduced density matrices pro- duced by all bipartite partitions. As implied by the full separability definition it is said to be fully m-partite sep- 36 Usually E is represented by symmetric adjacency matrix with elements Auv = Avu = 1 iff u = v are connected by the edge arable iff: 6 and Auv = 0 otherwise. Note, that there are no more than m(m 1) Ψ = ψ ψ . (84) − of the edges — pairs of vertices. | A1,...,Am i | A1 i⊗···⊗| Am i 2 32 the edges with a common vertex, the order of applying it one can use the symmetric Hilbert-Schmidt represen- unlock 1 4 3 4 the operations is arbitrary. Remarkably, the set of graph tation ̺ABCD = 4 (I⊗ + i=1 σi⊗ ). Thus, the state is states constructed is described by a polynomial number separable under any partition into two two-qubit parts. m(m 1)/2 of discrete parameters, while in general the Still, it is entangled underP any partition 1 versus 3 qubits set of− all states in the m-qubit Hilbert space is described since it violates PPT criterion with respect to this par- m unlock TA by an exponential 2 number of continuous parameters. tition ie. (̺ABCD) 0. This state has been shown m 6≥ Moreover, local unitary interconvertibility under i=1Ui to have applications in remote concentration of quan- of two graphs states is equivalent to convertibility⊗ under tum information (Murao and Vedral, 2001). The Smolin stochastic local operations and classical communications state has been also shown to be useful in reduction of (SLOCC). communication complexity via violation of Bell inequal- Any connected37 graph state G is fully entangled m- ities (Augusiak and Horodecki, 2006a). No bound en- particle state and violates some| Belli inequality. The lat- tangled states with fewer degrees of freedom useful for ter fact has been proven (G¨uhne et al., 2005) using alter- communication complexity are known so far. Gener- native (equivalent) description of graph states in terms alization of the state to multipartite case is possible of stabilizer formalism (Gottesman, 1997). Moreover the (Augusiak and Horodecki, 2006b; Wu et al., 2005). idea of efficiently locally measurable entanglement wit- A particularly interesting from the point of view of nesses that detect entanglement of graph states has been low partite case systems is a special class of partially proposed (Toth and G¨uhne, 2005a). This idea can be separable states called semiseparable. They are separable developed, to show that entanglement of any connected under all 1-(m-1) partitions : I1 = k , I2 = 1,...,k graph violates some Bell inequality that requires only 1, k +1,...,m , 1 k m{. It allows{ } to show{ a new− two measurements per each qubit site (Toth et al., 2006). type of entanglement:}} ≤ there≤ are semiseparable 3-qubit This is a very useful proposal of local detection of the states which are still entangled. To see it consider the mean value of corresponding entanglement witness. For next example (DiVincenzo et al., 2003a): of a review of many important and very interesting appli- Examples .- Consider the following product cations of graph states in quantum information, includ- states: 2 2 2 state composed on 3 parts ⊗ ⊗ ing details of their fundamental role in one-way quantum ABC generated by set defined as SShift = computing, see (Hein et al., 2005). 0 0 0 , + 1 , 1 + , + 1 , (with | i| i|=i |1 i|( 0i|−i |1 i|−i|)). Thisi |−i| seti| cani} be proved to |±i √2 | i ± | i define multipartite unextendible product basis in full B. Partial separability analogy to the bipartite case discussed in Sec. (VI.B.7): there is no product state orthogonal to subspace spanned Here we shall consider two other very important no- by SShift. Thus, in analogy to bipartite construction, tions of partial separability. The first one is just sep- the state ̺Shift = (I PShift)/4 where PShift projects arability with respect to partitions. In this case, the − onto the subspace spanned by SShift can be easily shown state of A ,...,A elementary subsystems is separable 1 m to be entangled as a whole (i.e. not fully separable) but with respect to a given partition I ,...,I , where I { 1 k} i PPT under all cuts (i.e. A BC, AB C, B AC). However are disjoint subsets of the set of indices I = 1,...,m | | | k N i i { } it happens that it is not only PPT but also separable ( l=1Il = I)iff ̺ = i=1 pi̺1 ̺k where any state under all cuts. This means that semiseparability is ̺∪i is defined on tensor product⊗···⊗ of all elementary Hilbert l P not equivalent to separability even in the most simple spaces corresponding to indices belonging to set Ii. Now, multipartite case like 3-qubit one. one may combine several separability conditions with re- Entanglement of semiseparable states shown here im- spect to several different partitions. This gives many mediately follows from construction by application of possible choices for partial separability. multipartite version of range criterion. It is also de- Let us show an interesting example of partial separa- tected by permutation criteria. Finally, one can use maps bility which requires even number of qubits in general. that are positive on all product states of two qubits but Example .- Consider four-qubit Smolin states (Smolin, not positive in general, described in Sec. VII.A (see 2001): (Horodecki et al., 2000e)). 4 1 Another interesting class is the set of U U U invari- ̺unlock = Ψi Ψi Ψi Ψi , (85) ant d d d states which comprises semiseparable⊗ ⊗ and ABCD 4 | ABih AB| ⊗ | CDih CD| ⊗ ⊗ i=1 fully 3-separable subclasses of states in one 5-parameter X family of states (see (Eggeling and Werner, 2001)). where Ψi are four Bell states. It happens that it is symmetrically| i invariant under any permutations (to see The moral of the story is that checking bipartite sep- arability with respect to all possible cuts in not enough to guarantee full separability. However separability with respect to some partial splittings gives still important 37 A graph state G is called connected if the corresponding graph generalization of separability and have interesting appli- G(V,E) is connected,| i that is when any two vertices from V are cations (see (D¨ur and Cirac, 2000a,b; D¨ur et al., 1999b; linked by the path of subsequent edges from E. Smolin, 2001) and Sec. XII). In this context we shall 33 describe below a particularly useful family of states. comprises separability conditions in terms of uncertainty Example .- Separability of the family of the states pre- relations that have been started to be developed first for sented here is fully determined by checking PPT cri- continuous variables (We must stress here that we do not terion under any possible partitions. To be more spe- consider entanglement measures which are also a non- cific, PPT condition with respect to some partition guar- linear functions of the state, but belong to a very spe- antees separability along that partition. The states cial class that has — in a sense — its own philosophy.). found some (D¨ur and Cirac, 2000b; D¨ur et al., 1999b) Such conditions will be described in subsections (VIII.A, important applications in activation of bound entangle- VIII.B) below. ment (D¨ur and Cirac, 2000a), nonadditivity of multipar- The second class of nonlinear separability conditions is tite quantum channels (D¨ur et al., 2004) and multipar- based on collective measurements on several copies and tite bound information phenomenon (Acin et al., 2004b). has attracted more and more attention recently. We shall This is the following m-qubit family (D¨ur et al., 1999b): present them in Sec. VIII.C.2. (m) a a a + + ̺ = λ Ψ Ψ + λ ( Ψ Ψ + Ψ− Ψ− ), 0 | 0ih 0| k | k ih k | | k ih k | a= k=0 X± X6 A. Uncertainty relation based separability tests (86) 1 where Ψ± = ( k1 k2 ... km 1 0 | k i √2 | i| i | − i| i ± The uncertainty relations have been first developed k1 k2 ... km 1 1 with ki = 0, 1, ki = ki 1 for continuous variables and applied to Gaussian states | i| i | − i| i m 1 ⊕ ≡ (ki + 1) mod 2 and k being one of 2 − real numbers (Duan et al., 2000) (see also (Mancini et al., 2002)). For defined by binary sequence k1,...,km 1. the bipartite case nonlinear inequalities for approxima- + − Let us put ∆ = λ0 λ0− 0 and define bipartite tions of finite dimensional Hilbert spaces in the limit of − ≥ splitting into two disjoint parts A(k) = subset with high dimensions have been exploited in terms of global { last (m-th) qubit , B(k) = subset without last qubit angular momentum-like uncertainties in (Kuzmich, 2000) } { with help of binary sequence k such that i-th qubit with further experimental application (Julsgaard et al., } belongs to A(k) (and not to B(k)) if and only if the 2001)38. sequence k1,...,km 1 contains ki = 0. Then one can General separability criteria based on uncertainty re- − (m) prove (D¨ur et al., 1999b) that (i) ̺ is separable with lation and valid both for discrete and continuous vari- respect to partition A(k), B(k) iff λ ∆/2 which { } k ≥ ables (CV) have been introduced in (Giovannetti et al., happens to be equivalent to PPT condition with respect 2003), and (Hofmann and Takeuchi, 2003) (the second T to that partition (ie. ̺ B 0) (ii) if PPT condition one was introduced for discrete variables but its general ≥ (m) is satisfied for all bipartite splittings then ̺ is fully formulation is valid also for the CV case). Soon fur- separable. Note that the condition (ii) above does not ther it has been shown (Hofmann, 2003) that PPT en- hold in general for other mixed states which can be seen tanglement can be detected by means of uncertainty rela- easily on 3-qubit semiseparable state ̺Shift recalled in tion introduced in (Hofmann and Takeuchi, 2003). This this section. That state is entangled but clearly satisfies approach has been further developed and simplified by PPT condition under all bipartite splittings. G¨uhne (G¨uhne, 2004) and developed also in entropic There is yet another classification that allows for strat- terms (G¨uhne and Lewenstein, 2004a). Another separa- ification of entanglement involved. For instance m- bility criterion in the two-mode continuous systems based particle system may be required to have at most s-particle on uncertainty relations with the particle number and the entanglement which means that it is a mixture of all destruction operators was presented (Toth et al., 2003), states such that each of them is separable with respect which may be used to detect entanglement in light field to some partition I ,...,I where all sets of indices { 1 k} or in Bose-Einstein condensates. Ik have cardinality not greater than s. All m-partite Let us recall briefly the key of the approach of lo- states that have at most m 1-particle entanglement − cal uncertainty relations (LUR) (Hofmann and Takeuchi, satisfy special Bell inequalities (see (Svetlichny, 1987) 2003) which has found a very nice application in the for bipartite and (Seevinck and Svetlichny, 2002) for gen- idea of macroscopic entanglement detection via magnetic eral case) and nonlinear separability criteria like that of susceptibility (Wie´sniak et al., 2005). Consider the set (Maassen and Uffink, 1988) which we shall pass to just N N of local observables Ai , Bi on Hilbert spaces in next section (see Sec. VIII.B). { }i=0 { }i=1 A, B respectively. Suppose that one has bounds on HsumH of local variances ie. δ(A )2 c , δ(B )2 i i ≥ a i i ≥ cb with some nonnegative values ca,cb and the vari- VIII. FURTHER IMPROVEMENTS OF ENTANGLEMENT 2 P 2 2 P ance definition δ(M) M ̺ M . Then for TESTS: NONLINEAR SEPARABILITY CRITERIA ̺ ≡ h i − h i̺ Then the nonlinear separability criteria into two dif- ferent classes. The first ones are based on functions of 38 The first use of uncertainty relation to detect entanglement (with results of few measurements performed in noncollective theoretically and experimentally) can be found in (Hald et al., manner (on one copy of ̺AB a time). This first time 1999) for spins of atomic ensembles. 34 any separable state ̺AB the following inequality holds An alternative interesting way is to consider an (Hofmann and Takeuchi, 2003): entropic version of uncertainty relations, as ini- tiated in (Giovannetti, 2004) and developed in δ(A I+I B )2 c + c . (87) (G¨uhne and Lewenstein, 2004a). This technique is a next i ⊗ ⊗ i ̺AB ≥ A B i step in application of entropies in detecting entangle- X ment. The main idea is (see (G¨uhne and Lewenstein, Note that by induction the above inequality can be ex- 2004a)) to prove that if sum of local Klein entropies41 tended to the multipartite case. Quite remarkably if the satisfies uncertainty relations S(A1) + S(A2) C, observables Ai and Bi are chosen in a special asym- ≥ S(B1)+ S(B2) C for systems A, B respectively then metric way, then the above inequality can be shown for any separable≥ state global Klein entropy must sat- (Hofmann, 2003) to detect entanglement of the family of isfy the same bound S(A1 B1) + S(A2 B2) C. (64) PPT states. The LUR approach has been general- This approach was also extended⊗ to multipartite⊗ ≥ case ized in (G¨uhne, 2004) to separability criteria via nonlocal (G¨uhne and Lewenstein, 2004a). uncertainty relations. That approach is based on the ob- servation that for any convex set (here we choose it to S be set of separable states), any set of observables Mi and B. Nonlinear improvement of entanglement witnesses the state ̺ = i pi̺i, ̺i the following inequality holds: ∈ S P In general any bound on nonlinear function of means δ(M )2 p δ(M )2 . (88) of observables that is satisfied by separable states but i ̺ ≥ k i ̺k violated by some entangled states can be in broad sense i k i X X X considered as “nonlinear entanglement witnesses”. Here It happens that in many cases it is relatively easy to show we shall consider those separability conditions that had that right hand side (RHS) is separated from zero only if their origin in entanglement witnesses (like for instance ̺ is separable, while at the same time LHS vanishes for Bell operators) and lead to nonlinear separability test. some entangled states 39. They are still all based on functions of mean values of Especially, as observed in (G¨uhne, 2004) if the observ- noncollective measurements (ie. single measurement per ables Mi have no common product eigenvector then the copy). 2 RHS must be strictly greater than zero since δ(M)Ψ van- Let us come back to more general nonlinear conditions ishes iff Ψ is an eigenvector of M. Consider now an inspired by Bell inequalities criteria. This interesting ap- arbitrary state that violates the range criterion in such proach has been first applied to the multipartite case a way that it has no product vector in its range R(̺) (Maassen and Uffink, 1988), where on the basis of Bell (an example is just the PPT entangled state produced inequalities nonlinear inequalities have been constructed. by UPB meted, but discrete value PPT entangled states These can discriminate between full m-particle entangle- from (Horodecki and Lewenstein, 2000). Now there is ment of the system A1...Am and the case when it contains a simple observation: any subspace ⊥ orthonormal to at most m 1-particle entanglement (cf. Sec. VII.B). the subspace having no product vectorH can be spanned It is worth− to note here that linear inequalities discrim- H r by (maybe nonorthogonal) entangled vectors Ψi i=1, inating between those two cases have been provided as r = dim . The technique of (G¨uhne, 2004){| is nowi} to Bell-like inequalities early in (Svetlichny, 1987) for 3 par- H take Mi = Ψi Ψi , i = 1,...,r and Mr+1 = PR(̺) ticles and extended in (Seevinck and Svetlichny, 2002) to where the last| oneih is| a projector onto the range of the m particles. Though they were not related directly to state, which — by definition — has no product state in entanglement, they automatically serve as an entangle- 40 the range. Note that since all Ψi belong to the kernel ment criteria since any separable state allows for LHVM of ̺ one immediately has LHS| ofi (88) vanishing. But as description of all local measurements, and as such satisfy one can see there is no product eigenvector that is com- all Bell inequalities. mon to all Mi-s hence as mentioned above, RHS must For small-dimensional systems, a nonlinear inequality be strictly positive which gives expected violation of the inspired by Bell inequality that is satisfied by all sepa- inequality (88). rable two-qubit states has been provided in (Yu et al., 2003). Namely the two-qubit state ̺ is separable if and only if the following inequality holds: 39 There is a more general form of this inequality in terms of co- 2 2 A1 B1 + A2 B2 + A3 I+I B3 variance matrices, which gives rise to new separability criteria; h ⊗ ⊗ i̺ h ⊗ ⊗ i̺ − a simple (strong) necessary and sufficient criterion for two-qubit q A3 B3 ̺ 1, (89) states was presented which violates LUR (G¨uhne et al., 2007). h ⊗ i ≤ (see also (Abascal and Bj¨ork, 2007) in this context) Other crite- rion for symmetric for n-qubit states have been presented in the form of a hierarchy of inseparability condition on the intergroup covariance matrices of even order (Devi et al., 2007). 41 The Klein entropy associated with an observable and a state is 40 By kernel of the state we mean the space spanned by vectors the classical entropy of probability distribution coming out of corresponding to zero eigenvalues of the state. measurement of this observable on the quantum state. 35 3 3 for all sets of dichotomic observables Ai i=1 Bi i=1 C. Detecting entanglement with collective measurements that correspond to two local bases orthonormal{ } { vectors} ai, bi that have the same orientation in Cartesian frame 1. Physical implementations of entanglement criteria with (the relation Ai = ai~σ,Bi = bi~σ holds). There is also a collective measurements link with PPT criterion: for any entangled state ̺ with Γ negative eigenvalue λmin of ̺ , the optimal setting of The idea of direct measurement of pure states entangle- observables makes LHS equal to 1 4λ . − max ment was considered first in (Sancho and Huelga, 2000) The above construction was inspired by a Bell-type and involved the first explicit application of collective entanglement witness (see (Yu et al., 2003) for details). measurements to entanglement detection (Acin et al., In this context a question arises if there is any system- 2000). In general the question here is how to detect atic way of nonlinear improvements of entanglement wit- entanglement physically by means of a little number of nesses. Here we shall describe a very nice nonlinear im- collective measurements that do not lead to complete to- provement that can be applied to any entanglement wit- mography of the state. Here we focus on the number ness (see (47)) and naturally exploits the isomorphism of estimated parameters (means of observables) and try (49). Before showing that, let us recall that the whole to diminish it. The fact that the mean of an observable entanglement witnesses formalism can be translated to may be interpreted as single binary estimated parame- the level of covariance matrix in continuous variables, ter (equivalent just to one-qubit polarization) has been and the nonlinear corrections to such a witnesses that proven in (Horodecki, 2003a; Paz and Roncaglia, 2003), are equivalent to some uncertainty relations can also be cf. (Brun, 2004). constructed (Hyllus and Eisert, 2006). The power of positive maps separability criteria The above announced (quadratic) nonlinear correc- and entanglement measures has motivated work on tion worked out in (G¨uhne and L¨utkenhaus, 2006a) in- implementations of separability criteria via collec- (h) (antih) volves a set of operators Xk = Xk + iXk and its tive measurements, introduced in (Horodecki, 2003b; mean Xk ̺, which (in practice) must be collected from Horodecki and Ekert, 2002) and significantly improved h i (h) in (Carteret, 2005; Horodecki et al., 2006f). On the mean values of their Hermitian (Xk ) and antihermitian (antih) other hand the entropic separability criteria has led to Xk parts respectively. The general form of the non- linear improvement of the witness W corresponds to the the separate notion of collective entanglement witnesses condition (Horodecki, 2003d) which will be described in more detail in the next section. = W α X 2 0, (90) The collective measurement evaluation of nonlinear F̺ h i̺ − k|h ki̺| ≥ k state functions (Ekert et al., 2002; Filip, 2002) (see also X (Fiurasek, 2002a; Horodecki, 2003d)) was implemented where real numbers αk and operators Xk are both cho- experimentally very recently in distance lab paradigm sen in such a way that for all possible separable states (Bovino et al., 2005). The method takes an especially ̺AB the condition ̺ 0 is satisfied. One can striking form in the two-qubit case, when not only unam- F AB ≥ see that the second term is a quadratic correction to biguous entanglement detection (Horodecki and Ekert, the original (linear) mean value of entanglement wit- 2002) but also estimation of such complicated entangle- ness. Higher order corrections are also possible though ment measure as entanglement of formation and Woot- they need not automatically guarantee stronger condition ters concurrence can be achieved by measuring only four (G¨uhne and L¨utkenhaus, 2006a). Let us illustrate this by collective observables (Horodecki, 2003b), much smaller Γ use of example of partial transpose map IA TB( ) = ( ) than 15 required by state estimation. The key idea ⊗ · · and the corresponding decomposable witness (see Eq. of the latter scheme is to measure four collective ob- (52)) coming from the minimal set of witnesses describ- servables A(2k) on 2k copies of the state that previ- ing set of states satisfying PPT conditions W = Ψ Ψ Γ. ously have been subjected to physical action of some | ih | In general, virtually any entanglement witness condition maps42. The mean values of these observables repro- can be improved in this way, though one has to involve duce all four moments A(2k) = λk of spectrum h i i i Hermitian conjugate of the map in Hilbert-Schmidt space λk of the square of the Wootters concurrence ma- and exploit the fact (see discussion in Sec. VI.B.3) that { } ˆ P trix C(̺) = √̺σ2 σ2̺∗σ2 σ2√̺. Note that, due single map condition is equivalent to continuous set of to the link (Wootters,⊗ 1998) between⊗ Wootters concur- entanglement witnesses conditions. Now, for the witness rence and entanglementp of formation, the latter can be Γ of the form W = Ψ Ψ one of the two versions are nat- also inferred in such an experiment. Recently, a col- ural. In the first| oneih chooses| single X = Φ Ψ (where | ih | lective observable very similar to those of (Horodecki, both vectors are normalized) and then single parameter 2003d), acting on two copies of quantum state which α is equal to the inverse of maximal Schmidt coefficient of Ψ . The second option is to choose Xi = Φ Ψi with Ψi |beingi an orthonormal basis in global Hilbert| ih space| HAB and then the choose αi = 1 for all parameters guarantees 42 They are so called physical structural approximations, which we positivity of the value (90). describe further in this section. 36 n detect two-qubit concurrence has been constructed and observable defined on ⊗ and measured on n copies n H implemented (Walborn et al., 2003). The observable is ̺⊗ of given state ̺. Also one defines the notion of n much simpler, however the method works under the mean of collective observable in single copy of state ̺⊗ (n) (n) n (n) promise that the state is pure. This approach can be as A := Tr(A ̺⊗ ). Now any observable W hh ii n also generalized to multiparty case using suitable factoris- defined on ( AB )⊗ that satisfies the following condition able observable corresponding to the concurrence (see H (n) (n) n (Aolita and Mintert, 2006)). W ̺sep := Tr(W ̺sep⊗ ) 0, (91) In methods involving positive maps criteria the main hh ii ≥ problem of how to physically implement unphysical maps but there exists an entangled state ̺ent such that has been overcome by the help of structural physical ap- (n) proximations (SPA) of unphysical maps. In fact for any W ̺ < 0, (92) hh ii ent Hermitian trace nonincreasing map L there is a prob- ability p such that the new map L˜ = p + (1 p)Λ is called collective (n-copy) entanglement witness. In can be physically implemented if is justD a fully− depo- (Horodecki, 2003d) collective entanglement witnesses re- larizing map that turns any stateD into maximally mixed producing differences of Tsallis entropies43, and as such one. Now one can apply the following procedure: (i) put verifying entropic inequalities with help of single observ- L =I Λ for some P but not CP map Λ, (ii) apply the able, have been designed. ⊗ new (physical) map L˜ to many copies of bipartite system Here we shall only recall the case of n = 2. The fol- in unknown state ̺, (iii) estimate the spectrum of the lowing joint observables Let us consider resulting state L˜(̺) = [I Λ](˜ ̺). Infer the spectrum of (2) ⊗ ′ ′ ′ L(̺)=[I Λ](̺) (which is an easy affine transformation X ′ ′ = VAA (VBB IBB ), AA BB ⊗ − ⊗ (2) of the measured spectrum of L˜(̺)) and check in this way ′ ′ ′ Y ′ ′ = (VAA IAA ) VBB . (93) whether the condition (42) for map Λ is violated. Gener- AA BB − ⊗ alization to multipartite systems is immediate. This test represent collective entanglement witnesses since they requires measurement of d2 observables instead of d4 1 − reproduce differences of Tsallis entropies with q = 2 ones needed to check the condition with prior state to- : X = S (̺ ) S (̺ ), Y = S (̺ ) hh ii 2 AB − 2 A hh ii 2 AB − mography. The corresponding quantum networks can be S2(̺B) which are always nonnegative for separable states easily generalized to multipartite maps criteria including (Horodecki et al., 1996c). realignment or linear contractions criteria (Horodecki, Collective (2-copy) entanglement witnesses were fur- 2003c). The implementation with help of local measure- ther applied in continuous variables for Gaussian states ments has also been developed (see (Alves et al., 2003). (Stobi´nska and W´odkiewicz, 2005). Recently it has been However, as pointed out by Carteret (Carteret, 2005), observed that there is a single 4-copy collective entangle- the disadvantage of the method is that SPA involved here ment witness that detects all (unknown) 2-qubit entan- requires in general significant amount of noise added to glement (Augusiak et al., 2006). On this basis a corre- the system. The improved method of noiseless detec- sponding universal quantum device, that can be inter- tion of PPT criterion, concurrence and tangle has been preted as a quantum computing device, has been de- worked out (Carteret, 2003, 2005) with help of polyno- signed. mial invariants technique which allows for very simple It is very interesting that the following collective en- and elegant quantum network designing. The problem tanglement witness (Mintert and Buchleitner, 2006): whether noiseless networks exist for all other positive maps (or contraction maps) have been solved quite re- (2) (+) ( ) ( ) ( ) W˜ ′ ′ =2P ′ P − ′ +2P − ′ P − ′ cently where general noiseless networks have been de- AA BB AA ⊗ BB AA ⊗ BB ( ) ( ) signed (Horodecki et al., 2006f). 4P − ′ P − ′ , (94) − AA ⊗ BB Finally let us note that the above techniques (+) ( ) have been also developed on the ground of continu- with P and P − being projectors onto symmetric ous variables (Fiurasek and Cerf, 2004; Pregnell, 2006; and antisymmetric subspace respectively (see VI.B.3), Stobi´nska and W´odkiewicz, 2005). has been shown to provide a lower bound on bipartite concurrence (̺ ) (Mintert et al., 2004) C AB (2) 2. Collective entanglement witnesses (̺ ) W˜ ′ ′ . (95) C AB ≥ −hh AA BB ii̺AB There is yet another technique introduced (Horodecki, ˜ (2) Note that WAA′BB′ shown above is just twice the sum of 2003d) that seems to be more and more important in the witnesses given in Eq. (93). context of experimental implementations. This is the notion of collective entanglement witness. Consider a bi- partite system AB on the space AB = A B. One H H ⊗ H (n) introduces here the notion of collective observable A 43 The family of Tsallis entropies parametrized by q > 0 is defined 1 q with respect to the single system Hilbert space as an as follows: Sq(ρ) = 1 q (Trρ 1). H − − 37 Further the technique leading to the above formula provide an algorithm deciding entanglement was based was applied in the construction of (single-copy) entan- on checking variational problem based on concurrence glement witnesses quantifying concurrence as follows vector (Audenaert et al., 2001c). The problem of ex- (Mintert, 2006). Suppose we have a given bipartite istence of classical algorithm that unavoidably identi- state σ for which we know the concurrence C(σ) ex- fies entanglement has been analyzed in (Doherty et al., actly. Then one can construct the following entangle- 2002, 2004) both theoretically and numerically and im- ment witness which is parametrised by state σ in a non- plemented by semidefinite programming methods. This linear way: W (σ ′ ′ ) = 4Tr ′ ′ (I σ ′ ′ V ′ approach is based on a theorem concerning the symmet- AB A B − A B ⊗ A B AA ⊗ VBB′ )/ (σA′B′ ) which has the following nice property: ric extensions of bipartite quantum state (Fannes et al., C 1988; Raggio and Werner, 1989). It has the following in- (̺AB) WAB(σA′B′ ) ̺ . (96) terpretation. For a given bipartite state ̺ one asks C ≥ −h i AB AB about the existence of a hierarchy of symmetric exten- sions, i.e. whether there exists a family of states ̺AB1...Bn 3. Detection of quantum entanglement as quantum computing (with n arbitrary high) such that ̺ABi = ̺AB for all with quantum data structure i = 1,...,n. It happens that the state ̺AB is separable if and only if such a hierarchy exists for each natural n It is interesting that entanglement detection schemes (see Sec. XVI). However, for any fixed n checking ex- involve schemes of quantum computing. The networks istence of such symmetric extension is equivalent to an detecting the moments of the spectrum of L(̺) that are instance of semidefinite programming. This leads to an elements of the entanglement detection scheme (Carteret, algorithm consisting in checking the above extendability 2005; Horodecki, 2003b,c; Horodecki and Ekert, 2002) for increasing n, which always stops if the initial state can be considered as having fully quantum input data ̺AB is entangled. However the algorithm never stops (copies of unknown quantum state) and give classical out- if the state is separable. Further another hierarchy has put — the moments of the spectrum. The most strik- been provided together with the corresponding algorithm ing network is a universal quantum device detecting 2- in (Eisert et al., 2004), extended to involve higher order qubit entanglement (Augusiak et al., 2006)44. This is a polynomial constraints and to address multipartite en- quantum computing unit with an input where (unknown) tanglement question. quantum data comes in. The input is represented by The idea of dual algorithm was provided in 4 copies of unknown 2-qubit state. Then they are cou- (Hulpke and Bruß, 2005), based on the observation that pled by a special unitary transformation (the network) to in checking separability of given state it is enough to a single qubit polarization of which is finally measured. consider countable set of product vectors spanning the The state is separable iff it is polarization is not less than range of the state. The constructed algorithm is dual to certain value. that described above, in the sense that its termination This rises the natural question of whether in the future is guaranteed iff the state is separable, otherwise it will it will be possible to design a quantum algorithm working not stop. It has been further realized that running both on fully quantum data (ie. copies of unknown multipar- the algorithms (ie. the one that always stops if the state tite states, entanglement of which has to be checked) that is entangled with the one that stops if the state is sep- will solve the separability/entanglement problem much arable) in parallel gives an algorithm that always stops faster than any classical algorithms (cf. next section). and decides entanglement definitely (Hulpke and Bruß, 2005). The complexity of both algorithms is exponential in IX. CLASSICAL ALGORITHMS DETECTING the size of the problem. It happens that it must be ENTANGLEMENT so. The milestone result that has been proved in a mean time was that solving separability problem is NP The first systematic methods of checking entangle- hard (Gurvits, 2002, 2003, 2004). Namely is it known ment of given state was worked out in terms of finding (Yudin and Nemirovskii, 1976) that if a largest ball con- the decomposition onto separable and entangled parts of tained in the convex set scales properly, and moreover the state (see (Lewenstein and Sanpera, 1998) and gen- there exists an efficient algorithm for deciding member- eralizations to case of PPT states (Kraus et al., 2000; ship, then one can efficiently minimize linear functionals Lewenstein et al., 2001)). The methods were based on over the convex set. Now, Gurvits has shown that for the systematic application of the range criterion involv- some entanglement witness optimization problem was in- ing however the difficult analytical part of finding prod- tractable. This, together with the results on radius of the uct states in the range of a matrix. A further attempt to ball contained within separable states (see Sec. X), shows that problem of separability cannot be efficiently solved. Recently the new algorithm via analysis of weak 44 It exploits especially the idea of measuring the mean of observ- membership problem as been developed together able by measurement of polarization of specially coupled qubit with analysis of NP hardness ((Ioannou, 2007; (Horodecki, 2003a; Paz and Roncaglia, 2003), cf. (Brun, 2004). Ioannou and Travaglione, 2006a; Ioannou et al., 2004)). 38 The goal of the algorithm is to solve what the authors maximally mixed subsystems) the singular values of the call “witness” problem. This is either (i) to write separa- correlation matrix T are invariants under such product ble decomposition up to given precision δ or (ii) to find an unitary transformations. The Euclidean lengths of the (according to slightly modified definition) entanglement real three-dimensional vectors with coordinates ri, sj witness that separates the state from a set of separable defined above are also similarly invariant. states by more than δ (the notion of the separation is Note, that a nice analogon of the tetrahedron in precisely defined). The analysis shows that one can find the state space for entangled two qudits was definedT and more and more precisely a likely entanglement witness investigated in the context of geometry of separability that detects the entanglement of the state (or find that (Baumgartner et al., 2007). It turns out that the analo- it is impossible) reducing the set of “good” (ie. possibly gon of the octahedron is no longer a polytope. detecting the state entanglement) witnesses by each step One can naturally ask about reasonable set of the pa- of the algorithm. The algorithm singles out a subroutine rameters or in general — functions of the state — that which in the standard picture (Horodecki et al., 1996a; are invariants of product unitary operations. Properly Terhal, 2000b) can be, to some extent, interpreted as an chosen invariants allow for characterization of local or- oracle calculating a “distance” of the given witness to the bits i.e. classes of states that are equivalent under lo- set of separable states. cal unitaries. (Note that any given orbit contains either Finally, note that there are also other proposals of al- only separable or only entangled states since entangle- gorithms deciding separability like (Zapatrin, 2005) (for ment property is preserved under local unitary product review see (Ioannou, 2007) and references therein). transformations). The problem of characterizing local orbits was analyzed in general in terms of polynomial invariants in (Grassl et al., 1998; Schlienz and Mahler, 1995). In case of two qubits this task was completed X. QUANTUM ENTANGLEMENT AND GEOMETRY explicitly with 18 invariants in which 9 are functionally independent (Makhlin, 2002) (cf. (Grassl et al., 1998)). Geometry of entangled and separable states Further this result has been generalized up to four is a wide branch of entanglement theory qubits (Briand et al., 2003; Luque and Thibon, 2003). ˙ (Bengtsson and Zyczkowski, 2006). The most simple Other way of characterizing entanglement in terms of and elementary example of geometrical representation local invariants was initiated in (Linden and Popescu, of separable and entangled states in three dimensions 1998; Linden et al., 1999b) by analysis of dimensional- is a representation of two-qubit state with maximally ity of local orbit. Full solution of this problem for mixed subsystems (Horodecki and Horodecki, 1996). mixed two-qubit states and general bipartite pure states Namely any two-qubit state can be represented in has been provided in (Ku´sand Zyczkowski,˙ 2001) and Hilbert-Schmidt basis σ σ where σ = I, and { i ⊗ j } 0 (Sino l¸ecka et al., 2002) respectively. For further develop- in this case the correlation matrix T with elements ment in this direction see (Grabowski et al., 2005) and t = Tr(̺σ σ ), i, j = 1, 2, 3 can be transformed ij i ⊗ j references therein. There are many other results con- by local unitary operations to the diagonal form. This cerning geometry or multiqubit states to mention only matrix completely characterizes the state iff local (Heydari, 2006; Levay, 2006; Miyake, 2003). density matrices are maximally mixed (which corre- There is another way to ask about geometrical proper- sponds to vanishing of the parameters r = Tr(σ I̺), i i ⊗ ties of entanglement. Namely to ask about volume of set sj = Tr(I σj ̺), for i, j = 1, 2, 3. It happens that after ⊗ 45 of separable states, its shape and the boundary of this set. diagonalizing , T is always a convex combination of The question about the volume of separable states was four matrices T0 = diag[1, 1, 1], T1 = diag[ 1, 1, 1], first considered in (Zyczkowski˙ et al., 1998) and extended T = diag[1, 1, 1], T = diag− [ 1, 1, 1] which− cor- 2 3 in (Zyczkowski,˙ 1999). In (Zyczkowski˙ et al., 1998) it has responds to maximally− mixed Bell− − basis.− This has a been proven with help of entanglement witnesses theory simple interpretation in threedimensional real space: a that for any finite dimensional system (bipartite or mul- tetrahedron with four vertices and coordinates corre- tipartite) the volume of separable states is nonzero. In sponding toT the diagonals above ((1, 1, 1) etc.). The particular there exists always a ball of separable states subset of separable states is an octahedron− that comes around maximally mixed state. An explicit bound on out from intersection of with its reflection around the ratio of volumes of the set of all states and that of the origin of the set of coordinates.T It is remarkable separable states S that all the states with maximally mixed subsystems are Ssep equivalent (up to product unitary operations UA UB) (d 1)(N 1) ⊗ vol( ) 1 − − to Bell diagonal states (a mixture of four Bell states S (97) (3)). Moreover, for all states (not only those with vol( sep) ≥ 1+ d/2 S for N partite systems each of dimension d was provided in (Vidal and Tarrach, 1999). This has inspired further 45 Diagonalizing matrix T corresponds to applying product UA UB discussion which has shown that experiments in NMR unitary operation to the state. ⊗ quantum computing may not correspond to real quan- 39 tum computing since they are performed on pseudop- that can be addressed in case of separable states. We ure states which are in fact separable (Braunstein et al., shall recall one of them: there is an interesting issue 1999). Interestingly, one can show (Kendon et al., 2002; of how probability of finding separable state (when the Zyczkowski˙ et al., 1998) that for any quantum system on probability is measured up an a priori probability mea- some Hilbert space maximal ball inscribed into a set sure µ) is related to the probability (calculated by in- of mixed states is locatedH around maximally mixed states duced measure) of finding a random boundary state to 2 1 be separable. The answer of course will depend on a and is given by the condition R(̺) = Tr(̺ ) dim 2 1 . Since this condition guarantees also the positivity≥ H − of choice of probability measure, which is by no means any unit trace operator, and since, for bipartite states unique. Numerical analysis suggested (Slater, 2005a,b) 2 Γ 2 Tr(̺AB) = Tr[(̺AB ) ] this means that the maximal ball that in two-qubit case the ratio of those two probabil- contains PPT states (the same argument works also for ities is equal to two if one assumes measure based the multipartite states (Kendon et al., 2002)). In case of 2 2 Hilbert-Schmidt distance. Recently it has been proven ⊗ or 2 3 this implies also separability giving a way to es- that for any dA dB system this rate is indeed 2 if we ⊗ timate⊗ volume of separable states from below. ask about set of PPT states rather than separable ones (Szarek et al., 2006). For 2 2 and 2 3 case this repro- These estimates have been generalized to multipartite ⊗ ⊗ states (Braunstein et al., 1999) and further much im- duces the previous conjecture since PPT condition char- proved providing very strong upper and lower bounds acterizes separability there (see Sec. VI.B.2). Moreover, with help of a subtle technique exploiting among oth- it has been proven (see (Szarek et al., 2006) for details) ers entanglement witnesses theory (Gurvits and Barnum, that bipartite PPT states can be decomposed into the so 2002, 2003, 2005). In particular it was shown that for bi- called pyramids of constant height. partite states, the largest separable ball around mixed states. One of applications of the largest separable ball results is the proof of NP-hardness of deciding weather a XI. THE PARADIGM OF LOCAL OPERATIONS AND state is separable or not (Gurvits, 2003) (see Sec. IX). CLASSICAL COMMUNICATION (LOCC) There is yet another related question: one can de- A. Quantum channel — the main notion fine the state (bipartite or multipartite) ̺ that re- mains separable under action of any unitary opera- Here we shall recall that the most general quantum tion U. Such states are called absolutely separable operation that transforms one quantum state into the (Ku´sand Zyczkowski,˙ 2001). In full analogy one can de- other is a probabilistic or stochastic physical operation of fine what we call here absolute PPT property (ie. PPT the type property that is preserved under any unitary transfor- mation). The question of which states are absolutely ̺ Λ(̺)/Tr(Λ(̺)), (98) PPT has been fully solved for 2 n systems (Hildebrand, → ⊗ 2005): those are all states spectrum of which satisfies with trace nonincreasing CP map, i.e. a map satisfying 2n the inequality: λ1 λ2n 1 + λ2n 2λ2n where λi i=1 Tr(Λ(̺)) 1 for any state ̺, which can be expressed in ≤ − − { } are eigenvalues of ̺ in decreasing order. This immedi- the form ≤ ately provides the characterizationp of absolutely separa- ble states in dA dB systems with dAdB 6 since PPT Λ(̺)= V (̺)V †, (99) ⊗ ≤ i i is equivalent to separability in those cases. Note that for i 2 2 states this characterization has been proven much X ⊗ earlier by different methods (Verstraete et al., 2001b). In with i Vi†Vi I (domain and codomain of operators Vi particular it follows that for those low dimensional cases called Kraus operators≤ ((Kraus, 1983)) are in general dif- the set of absolutely separable states is strictly larger ferent).P The operation above takes place with the prob- than that of maximal ball inscribed into the set of all ability Tr(Λ(̺)) which depends on the argument ̺. The states. Whether it is true in higher dimensions remains probability is equal to one if an only if the CP map Λ an open problem. is tracepreserving (which corresponds to i Vi†Vi = I in Speaking about geometry of separable states one can (99); in such a case Λ is called deterministic or a quantum not avoid a question about what is a boundary ∂ channel. P of set of states? This, in general not easy ques-S tion, can be answered analytically in case of two-qubit case when it can be shown to be smooth (Djokovic, B. LOCC operations 2006) relatively to set of all 2-qubit states which is closely related to the separability characterization We already know that in the quantum teleportation Γ (Augusiak et al., 2006) det(̺AB) 0. Interestingly, process Alice performs a local measurement with max- ≥ i it has been shown, that the set of separable state is imally entangled projectors PAA′ on her particles AA′ not polytope (Ioannou and Travaglione, 2006b) it has no and then sends classical information to Bob (see Sec i faces (G¨uhne and L¨utkenhaus, 2006b). III.C). Bob performs accordingly a local operation UB There are many other interesting geometrical issues on his particle B. Note that the total operation acts 40 ′ i i i i on ρ as: ΛAA B(ρ) = i PAA′ UB(ρ)PAA′ (UB)†. operation is ̺ ̺ ̺P P T i. e. the process of adding This operation belongs to so called⊗ one-way LOCC⊗ class some PPT state.→ ⊗ which is very importantP in quantum communication the- There is an order of inclusions C1 C2a, C2b C3 ory. The general LOCC paradigm was formulated in C4 C5, where all inclusions are strict⊂ ie. are not⊂ equal-⊂ (Bennett et al., 1996d). ities.⊂ The most intriguing is nonequivalence C3 = C4 In this paradigm all what the distant parties (Alice and which follows from so called nonlocality without entangle-6 Bob) are allowed is to perform arbitrary local quantum ment (Bennett et al., 1999a): there are examples of prod- operations and sending classical information. No transfer uct basis which are orthonormal (and hence perfectly dis- of quantum systems between the labs is allowed. It is a tinguishable by suitable von Neumann measurement) but natural class for considering entanglement processing be- are not products of two orthonormal local ones which rep- cause classical bits cannot convey quantum information resent vectors that cannot be perfectly distinguished by and cannot create entanglement so that entanglement re- parties that are far apart and can use only LOCC op- mains a resource that can be only manipulated. More- erations. Let us stress, that the inclusion C3 C4 is over one can easily imagine that sending classical bits is extensively used in context of LOCC operations.⊂ This is much more cheaper than sending quantum bits, because because they are hard to deal with, as characterized in a it is easy to amplify classical information. Sometimes it difficult way. If instead one deals with separable or PPT is convenient to put some restrictions also onto classi- operations, thanks to the inclusion, one can still conclude cal information. One then distinguishes in general the about LOCC ones. following subclasses of operations described below. We Subsequently if it is not specified, the term LOCC will assume all the operations (except of the local operations) be referred to the most general class of operations with to be trace nonincreasing and compute the state transfor- help of local operations and classical communication — mation as in (98) since the transformation may be either namely the class C3 above. stochastic or deterministic (ie. quantum channel). Below we shall provide examples of some LOCC oper- C1 - class of local operations .- In this case no commu- ations nication between Alice and Bob is allowed. The math- Example .- The (deterministic) “U U, U U ∗ ⊗ ⊗ ematical structure of the map is elementary: ΛAB∅ = twirling” operations: ΛA ΛB with ΛA,ΛB being both quantum channels. As we said⊗ already this operation is always deterministic. τ(̺)= dUU U̺(U U)†, C2a - class of “one-way” forward LOCC operations .- ⊗ ⊗ Here classical communication from Alice to Bob is al- Z i τ ′(̺)= dUU U ∗̺(U U ∗)†. (101) lowed. The form of the map is: ΛAB→ (̺) = i VA i i ⊗i ⊗ ⊗ I ([I Λ ](̺))(V )† I with deterministic maps Λ Z B A ⊗ B A ⊗ B B which reflect the fact that Bob is not allowed toP perform Here dU is a uniform probabilistic distribution on set of “truly stochastic” operation since he cannot tell Alice unitary matrices. They are of “one-way” type and can whether it has taken place or not (which would happen be performed in the following manner: Alice pick up ran- only with some probability in general ). domly the operation U rotates her subsystem with it and C2b - class of “one-way” backward LOCC operations .- sends the information to Bob which U she had chosen. i i i Here one has Λ← (̺)= IA V [Λ IB](̺)IA (V )†. Bob performs on his side either U or U (depending on AB i ⊗ B A⊗ ⊗ B ∗ The situation is the same as in C2a but with the roles of which of the two operations they wanted to perform). Alice and Bob interchanged.P The integration above can be made discrete which fol- C3 - class of “two-way” classical communication .- lows from Caratheodory’s theorem. Here both parties are allowed to send classical commu- The very important issue is that any state can be de- nication to each other. The mathematical form of the polarized with help of τ, τ ′ to Werner (71) and isotropic operation is quite complicated and the reader is referred (72) state respectively. This element is crucial for entan- to (Donald et al., 2002). Fortunately, there are other two glement distillation recurrence protocol (Bennett et al., larger in a sense of inclusion classes, that are much more 1996c) (see Sec. XII.B). easy to deal with: the classes of separable and PPT op- Another example concerns local filtering erations. (Bennett et al., 1996b; Gisin, 1996b): C4 - Class of separable operations.- This class was con- Example .- The (stochastic) local filtering operation is sidered in (Rains, 1997; Vedral and Plenio, 1998). These are operations with product Kraus operators: A B̺A† B† LOCCfilter(̺)= ⊗ ⊗ , sep L Tr(A B̺A† B†) Λ (̺)= Ai Bi̺A† B†, (100) ⊗ ⊗ AB ⊗ i ⊗ i i A†A B†B I I. (102) X ⊗ ≤ ⊗ which satisfy i Ai†Ai Bi†Bi =I I. This operation requires two-way communication (each C5. PPT operations⊗ .- Those are⊗ operations (Rains, party must send a single bit), however if A = I (B = I) 1999, 2001) ΛP P T such that (ΛP P T [( )Γ])Γ is completely then it becomes one-way forward (backward) filtering. positive. We shall see that the simplest· example of such This one-way filtering is a crucial element of a universal 41 protocol of entanglement distillation (Horodecki et al., 1996c)) there was presented a protocol for two qubit 1997) (see Sec. XII.D). states which originates from cryptographic privacy am- The local filtering operation is special case of a more plification protocol, called hashing. Following this work general class operations called stochastic separable op- we consider here the so called Bell diagonal states which erations, which includes class C4. They are defined as are mixtures of two qubit Bell basis states (3). Bell di- follows (Rains, 1997; Vedral and Plenio, 1998): agonal states ρBdiag are naturally parametrized by the probability of mixing p . For these states, the one-way A B ̺A† B† { } Λsep (̺)= i i ⊗ i i ⊗ i (103) hashing protocol yields singlets at a rate 1 H(p), thus AB proving46 E (ρ ) 1 H(p) In two-qubit− case there Pi TrAi†Ai Bi†Bi̺ D Bdiag ⊗ are four Bell states (3).≥ The−n copies of the two qubit Bell where A†Ai B†BPi I I. These operations can be diagonal state ρ can be viewed as a classical mix- i i ⊗ i ≤ ⊗ Bdiag applied probabilistically via LOCC operations. ture of strings of n Bell states. Typically, there are only P nH( p ) about 2 { } of such strings that are likely to occur (Cover and Thomas, 1991). Since the distillation proce- XII. DISTILLATION AND BOUND ENTANGLEMENT dure yields some (shorter) string of singlets solely, there is a straightforward “classical” idea, that to distill one Many basic effects in quantum information theory needs to know what string of Bells occurred. This knowl- based exploit pure maximally entangled state ψ+ . How- | i edge is sufficient as one can then rotate each non-singlet ever, in laboratories we usually have mixed states due to Bell state into singlet state easily as in a dense coding imperfection of operations and decoherence. A natural protocol (see sec III). question arises, how to deal with a noise so that one could Let us note, that sharing φ instead of φ+ can be take advantages of the interesting features of pure entan- − viewed as sharing φ+ with a phase error. Similarly ψ+ gled states. This problem has been first considered by means bit error and ψ – both bit and phase error. The Bennett, Brassard, Popescu, Schumacher, Smolin, and − identification of the string of all Bell states that have Wootters in 1996 (Bennett et al., 1996c). In their sem- occurred is then equivalent to learning which types of inal paper, they have established a paradigm for purifi- errors occurred in which places. Thus the above method cation (or distillation) of entanglement. When two dis- can be viewed as error correction procedure47. tant parties share n copies of a bipartite mixed state ρ, which contain noisy entanglement, they can perform Now, as it is well known, to distinguish between nH( p ) nH( p ) 2 { } strings, one needs log 2 { } = nH( p ) bi- some LOCC and obtain in turn some (less) number of k { } copies of systems in state close to a singlet state which nary questions. Such a binary question can be the fol- contains pure entanglement. A sequence of LOCC op- lowing: what is the sum of the bit-values of the string erations achieving this task is called entanglement pu- at certain i1,...,ik positions, taken modulo 2? In other rification or entanglement distillation protocol. We are words: what is the parity of the given subset of positions interested in optimal entanglement distillation protocols in the string. From probabilistic considerations it follows, i.e. those which result in maximal ratio k in limit of that after r such questions about random subset of posi- n tions (i.e. taking each time random k with 1 k 2n) large number n of input copies. This optimal ratio is ≤ ≤ called distillable entanglement and denoted as E (see the probability of distinguishing two distinct strings is no D r less than 1 2− , hence the procedure is efficient. Sec. XV for formal definition). Having performed entan- − glement distillation, the parties can use obtained nearly The trick of the “hashing” protocol is that it can be singlet states to perform quantum teleportation, entan- done in quantum setting. There are certain local unitary glement based quantum cryptography and other useful operations for the two parties, so that they are able to pure entanglement based protocols. Therefore, entangle- collect the parity of the subset of Bell states onto a single ment distillation is one of the fundamental concepts in Bell state and then get to know it locally measuring this dealing with quantum information and entanglement in Bell state and comparing the results. Since each answer general. In this section we present the most important to binary question consumes one Bell state, and there entanglement distillation protocols. We then discuss the are nH( p ) questions to be asked one needs at least { } possibility of entanglement distillation in general and re- H( p ) < 1 to obtain nonzero amount of not measured { } port the bound entangled states which being entangled Bell states. If this is satisfied, after the protocol, there can not be distilled. are n nH( p ) unmeasured Bell states in a known state. − { } A. One-way hashing distillation protocol 46 It is known, that if there are only two Bell states in mixture, If only one party can tell the other party her/his result then one-way hashing is optimal so that distillable entanglement during the protocol of distillation, the protocol is called is equal 1 H(p) in this case. 47 Actually this− reflects a deep relation developed in (Bennett et al., one-way, and two-way otherwise. One-way protocols are 1996d) between entanglement distillation and the large domain closely connected to error correction, as we will see be- of quantum error correction designed for quantum computation low. In (Bennett et al., 1996d) (see also (Bennett et al., in presence of noise. 42 The parties can then rotate them all to a singlet form (VI.B.9). In one step of the above recurrence procedure (that is correct the bit and phase errors), and hence distill this parameter improves with respect to the preceding singlets at an announced rate 1 H( p ). one according to the rule: This protocol can be applied− even{ if} Alice and Bob 2 1 2 share a non Bell diagonal state, as they can twirl the state F + 9 (1 F ) F ′(F )= − . (104) applying at random one of the four operations: σx σx, F 2 + 2 F (1 F )+ 5 (1 F )2 σ σ , σ σ , I I (which can be done one two-way).⊗ 3 − 9 − y ⊗ y z ⊗ z ⊗ The resulting state is a Bell diagonal state. Of course 1 Now, if only F > 2 , then the above recursive map con- this operation often will kill entanglement. We will see verges to 1 for sufficiently big initial number of copies. how to improve this in Secs. XII.B and XII.D. The idea behind the protocol is the following. Step The above idea has been further generalized leading to 1 decreases bit error (i.e. in the mixture the weight of general one-way hashing protocol which is discussed in correlated states φ± increases). At the same time, the Sec. XII.F. phase error increases,| i i.e. the bias between states of type + and those of type gets smaller. Then there is twirling step, which equalizes− bit and phase error. Provided that B. Two-way recurrence distillation protocol bit error went down more than phase error went up, the net effect is positive. Instead of twirling one can apply de- The hashing protocol cannot distill all entangled Bell- terministic transformation (Deutsch et al., 1996), which diagonal states (one easily find this, knowing that those is much more efficient. states are entangled if and only if some eigenvalue is greater than 1/2). To cover all entangled Bell diagonal states one can first launch a two-way distillation proto- col to enter the regime where one-way hashing protocol C. Development of distillation protocols — bipartite and multipartite case works. The first such protocol, called recurrence, was announced already in the very first paper on distillation (Bennett et al., 1996c), and developed in (Bennett et al., The idea of recurrence protocol was developed in dif- 1996d). It works for two qubit states satisfying F = ferent ways. The CNOT operation, that is made in Trρ φ+ φ+ > 1 with φ+ = 1 ( 00 + 11 ). step 1.1 of the above original protocol, can be viewed | ih | 2 | i √2 | i | i as a permutation. If one apply some other permuta- The protocol is defined by application of certain re- tion acting locally on n 2 qubits and performs a cursive procedure. The procedure is probabilistic so that testing measurements of steps≥ 1.2 1.3 on m 1 one it may fail however with small probability. Moreover it obtains a natural generalization of− this scheme,≥ devel- consumes a lot of resources. In each step the procedure oped in (Dehaene et al., 2003) for the case of two-qubit uses half of initial number of states. states. It follows that in case of n = 4,m = 1, and a special permutation this protocol yields higher distilla- 1. Divide all systems into pairs: ρAB ρA′B′ with AB ⊗ tion rate. This paradigm has been further analyzed in being source system and A′B′ being a target one. For each such pair do: context of so called code based entanglement distillation protocols (Ambainis and Gottesman, 2003; Matsumoto, 1.1 Apply CNOT transformation with control 2003) in (Hostens et al., 2004). The original idea of at A(B) system and target at A′(B′) for (Bennett et al., 1996c) linking entanglement distillation Alice (Bob). protocols and error correction procedures (Gottesman, 1.2 Measure the target system in computa- 1997) have been also developed in context of quantum tional basis on both A′ and B′ and com- key distribution. See in this context (Ambainis et al., pare the results. 2002; Gottesman and Lo, 2003) and discussion in section 1.3 If the results are the same, keep the AB XIX.A. The original recurrence protocol was generalized to system and discard the A′B′. Else remove both systems. higher dimensional systems in two ways in (Alber et al., 2001b; Horodecki and Horodecki, 1999). An inter- 2. If no system survived, then stop — the algorithm esting improvement of distillation techniques due to failed. Else twirl all the survived systems (Vollbrecht and Verstraete, 2005) where a protocol that turning them into Werner states. interpolates between hashing and recurrence one was provided. This idea has been recently developed in 2.1 For one of the system do: if its state ρ′ (Hostens et al., 2006a). Also, distillation was consid- satisfies 1 S(ρ′) > 0 then stop (hashing protocol will− work) else go to step (1). ered in the context of topological quantum memory (Bombin and Martin-Delgado, 2006). In the above procedure, one deals with Werner states, The above protocols for distillation of bipartite entan- because of twirling in step 2. In two-qubit case, Werner glement can be used for distillation of a multipartite en- states are equivalent to isotropic states and hence are tanglement, when n parties are provided many copies of parameterized only by the singlet fraction F (See Sec. a multipartite state. Namely any pair of the parties can 43 distill some EPR pairs and then, using e.g. teleporta- D. All two-qubit entangled states are distillable tion, the whole group can redistribute a desired multi- partite state. Advantage of such approach is that it is The recurrence protocol, followed by hashing can dis- 1 independent from the target state. In (D¨ur and Cirac, till entanglement only from states satisfying F > 2 . One 2000b; D¨ur et al., 1999b) a sufficient condition for distil- can then ask what is known in the general case. In this lability of arbitrary entangled state from n-qubit multi- section we present the protocol which allows to overcome partite state has been provided, basing on this idea. this bound in the case of two qubits. The idea is that with certain (perhaps small) probability, one can conclu- However, as it was found by Murao et. al in the first sively transform a given state into a more desired one, so paper (Murao et al., 1998) on multipartite entanglement that one knows if the transformation succeeded. There distillation, the efficiency of protocol which uses bipartite is an operation which can increase the parameter F , so entanglement distillation is in general less then that of di- that one can then perform recurrence and hashing proto- rect distillation. The direct procedure is presented there col. Selecting successful cases in the probabilistic trans- which is a generalization of bipartite recurrence protocol formation is called local filtering (Gisin, 1996b), which of n-partite GHZ state from its “noisy” version (mixed gives the name of the protocol. The composition of fil- with identity). In (Maneva and Smolin, 2002) the bipar- tering, recurrence and hashing proves the following result tite hashing protocol has been generalized for distillation (Horodecki et al., 1997): of n-partite GHZ state. Any two-qubit state is distillable if and only if it is • entangled. In multipartite scenario, there is no distinguished state like a singlet state, which can be a universal target state Formally, in filtering protocol Alice and Bob are given in entanglement distillation procedures. There are how- n copies of state ρ. They then act on each copy with ever some natural classes of interesting target states, in- an operation given by Kraus operators (see note in Sec. cluding the commonly studied GHZ state. An exem- VI) A B, √I A A B, A √I B B, √I A A plary is the class of the graph states (see Sec. VII.A), † † † √ { ⊗ − ⊗ ⊗ − − ⊗ related to one-way quantum computation model. A class I B†B . The event corresponding to Kraus operator A −B is the} desired one, and the complement corresponds of multiparticle entanglement purification protocols that ⊗ allow for distillation of these states was first studied in to failure in improving properties of ρ. In the case of (D¨ur et al., 2003) where it is shown again to outperform success the state can be transformed into bipartite entanglement purification protocols. It was fur- A BρA† B† ther developed in (Aschauer et al., 2005) for the subclass σ = ⊗ ⊗ , (105) Tr(A BρA B ) of graph states called two-colorable graph states. The ⊗ † ⊗ † recurrence and breeding protocol which distills all graph however only with probability p = Tr(A BρA B ). states has been also recently found (Kruszy´nska et al., † † After having performed the operation, Alice⊗ and Bob⊗ se- 2006) (for a distillation of graph states under local noise lect all good events and keep them, while removing the see (Kay et al., 2006)). copies, for which they failed. Suppose now that Alice and Bob are given n copies The class of two-colorable graph states is locally equiv- of entangled state ρ having F 1 . They would like to ≤ 2 alent to the class of so called CSS states that stems from obtain some number k of states with F > 1 . We show the quantum error correction (Calderbank and Shor, 2 now how one can build up a filtering operation which 1996; Steane, 1996a,b). The distillation of CSS states does this task. Namely any two-qubit entangled state has been studied in context of multipartite quantum becomes not positive after partial transpose (is NPT). cryptographic protocols in (Chen and Lo, 2004) (see Sec. Then, there is a pure state ψ = a ij such, that XIX.E.3). Recently, the protocol which is direct gener- | i ij ij | i alization of original hashing method (see Sec. XII.A) ψ ρΓ ψ < P0, (106) has been found, that distills CSS states (Hostens et al., h | AB| i 2006b). This protocol outperforms all previous versions with Γ denoting partial transpose on Bob’s subsystem. of hashing of CSS states (or their subclasses such as Bell It can be shown, that a filter A B = M I with diagonal states) (Aschauer et al., 2005; Chen and Lo, ψ [M ] = a is the needed one. That⊗ is the state⊗σ given 2004; D¨ur et al., 2003; Maneva and Smolin, 2002). Dis- ψ ij ij by tillation of the state W which is not a CSS state, has been studied recently in (Miyake and Briegel, 2005). In Mψ IρABM † I (Glancy et al., 2006) a protocol of distillation of all stabi- σ = ⊗ ψ ⊗ , (107) lizer states (a class which includes CSS states) based on Tr(Mψ IρABM † I) ⊗ ψ ⊗ stabilizer codes (Gottesman, 1997; Nielsen and Chuang, 1 2000) was proposed. Based on this, a breeding proto- which is the result of filtering, fulfills F (σ) > 2 . col for stabilizer states was provided in (Hostens et al., The filtering distillation protocol is just this: Alice ap- 2006c). plies to each state a POVM defined by Kraus operators 44 coh Mψ I, [ I M † Mψ] I . She then tells Bob when she ent information defined as IA B = S(ρB) S(ρAB) { ⊗ − ψ ⊗ } i − ≡ succeeded.q They select in turn on average np pairs with S(A B). − The| original proof of hashing inequality p = Tr(Mψ† Mψ IρAB). They then launch recurrence and hashing protocols⊗ to distill entanglement, which yields (Devetak and Winter, 2004b, 2005) was based on nonzero distillable entanglement. cryptographic techniques, where one first performs Similar argument along these lines, gives generaliza- error correction (corresponding to correcting bit) and tion of two-qubit distillability to the case of NPT states then privacy amplification (corresponding to correcting acting on 2 N Hilbert space (D¨ur and Cirac, 2000a; phase), both procedures by means of random codes (we C ⊗C will discuss it in Sec. XIX). D¨ur et al., 2000a). It follows also, that for N = 3 all coh Another protocol that distills the amount IA B of sin- states are also distillable if and only if they are entan- i gled, since any state on 2 3 is entangled if and only glets from a given state is the following (Horodecki et al., if it is NPT. The sameC equivalence⊗C has been shown for 2005h, 2006e): given many copies of the state, Al- all rank two bipartite states (Horodecki et al., 2006f) ice projects her system onto so-called typical subspace (Schumacher, 1995a) (the probability of failure is expo- nentially small with number of copies). Subsequently, E. Reduction criterion and distillability she performs incomplete measurement Pi where the { n(}SB SAB ) projectors Pi project onto blocks of size 2 − . If One can generalize the filtering protocol by use of re- the measurement is chosen at random according to uni- duction criterion of separability (see Sec. VI). It has been form measure (Haar measure), it turns out that for any shown that any state that violates reduction criterion is given outcome Alice and Bob share almost maximally distillable (Horodecki and Horodecki, 1999). Namely, for entangled state, hence equivalent to n(S S ) e-bits. B − AB states which violate this criterion, there exists filter, such Of course, for each particular outcome i the state is dif- 1 that the output state has fidelity F > d , where F is over- ferent, therefore one-way communication is needed (Bob + 1 d 1 lap with maximally entangled state Φ = − ii . has to know the outcome i). | i √d i=0 | i The above bound is often too rough (e.g. because one Such states is distillable, similarly, as for two-qubit states 1 P can distill states with negative coherent information us- with F > 2 . The simplest protocol (Braunstein et al., coh ing recurrence protocol). Coherent information IA B is 1999) is the following: one projects such state using local i rank two projectors P = 0 0 + 1 1 , and finds that not an entanglement monotone. It is then known, that | ih | 1| ih | in general the optimal protocol of distillation of entan- obtained two-qubit state has F > 2 , hence is distillable. coh The importance of this property of reduction criterion glement is some two-way protocol which increases I , lies in the fact, that its generalization to continuous vari- followed by general hashing protocol: That is we have ables allowed to show that all two-mode Gaussian states (Horodecki et al., 2000c) which violate PPT criterion are distillable. coh ED(ρ) = sup I (Λ(ρ)). (109) Λ LOCC ∈ F. General one-way hashing It is however not known how to attain the highest coher- ent information via two-way distillation protocol. One can ask what is the maximal yield of singlet as a function of a bipartite state ρAB that can be ob- tained by means of one way classical communication. G. Bound entanglement — when distillability fails In this section we will discuss protocol which achieves this task (Devetak and Winter, 2005; Horodecki et al., Since the seminal paper on distillation (Bennett et al., 2005h, 2006e). 1996c), there was a common expectation, that all entan- In Sec. (XII.A) we learned a protocol called one-way gled bipartite states are distillable. Surprisingly it is not hashing which for Bell diagonal states with spectrum the case. It was shown in (Horodecki et al., 1998a) that given by distribution p , gives 1 H( p ) of distillable the PPT states cannot be distilled. It is rather obvi- entanglement. Since in{ case} of these− states,{ } the von Neu- ous, that one cannot distill from separable states. Inter- B mann entropy of subsystem reads S(ρBdiag) = 1 and the estingly, the first example of entangled PPT state had AB been already known from (Horodecki, 1997). Generally total entropy of the state is equal to S(ρBdiag)= H( p ), there has been a common believe that in general there{ } the states that are entangled yet not distillable are called should be a one-way protocol that yields bound entangled. It is not known if there are other bound entangled states than PPT entangled states. There are ED(ρAB) S(ρB) S(ρAB). (108) quite many interesting formal approaches allowing to ob- ≥ − tain families of PPT entangled state. There is however This conjecture has been proved by Devetak and Winter no operational, intuitive “reason” for existence of this (Devetak and Winter, 2005). mysterious type of entanglement. The above inequality, called hashing inequality, states Let us first comment on the non-distillability of separa- that distillable entanglement is lower bounded by coher- ble states. The intuition for this is straightforward: sep- 45 arable states can be created via LOCC from scratch. If entangled state can be filtered, to such a state, that af- one could distill singlets from them, it would be creating ter proper twirling, one obtains entangled Werner state. something out of nothing. This reasoning of course does The question is hard to answer because of its asymptotic not hold for entangled PPT states. However, one can nature. A necessary and sufficient condition for entangle- look from a different angle to find relevant formal similar- ment distillation (Horodecki et al., 1998a) can be stated ities between PPT states and separable states. Namely, as follows: concerning separable states one can observe, that the fi- + + 1 delity F = Trσsep Φ Φ is no greater then for σsep Bipartite state ρ is distillable if and only if there d • n d d | ih | exists n such, that ρ is n-copy distillable (i.e. ρ⊗ acting on . Since LOCC operations can only trans- 49 form separableC ⊗C state into another separable state, (i.e. can be filtered to a two-qubit entangled state) . the set of separable states is closed under LOCC oper- for some n. It is however known, that there ations), one cannot distill singlet from separable states are states which are not n-copy distillable but they since one cannot increase the singlet fraction. are n + 1-copy distillable (Watrous, 2003) (see also It turns out that also PPT states do not admit higher (Bandyopadhyay and Roychowdhury, 2003)), for this fidelity than 1/d as well as are closed under LOCC op- reason, the numerical search concerning distillability bas- erations. Indeed we have Trρ Φ+ Φ+ = 1 TrρΓ V AB| ih | d AB ing on few copies may be misleading, and demanding in which can not exceed 1 (here V is the swap operator d computing resources. There is an interesting character- (48)). Indeed, ρΓ 0, so that TrρΓ V can be viewed AB ≥ AB ization of n copy distillable states in terms of entangle- as an average value of a random variable which can not ment witnesses found in (Kraus et al., 2002). Namely, a exceed 1 since V has eigenvalues 1 (Rains, 1999, 2001). state is separable iff the following operator To see the second feature, note that± any LOCC operation + n Γ Λ acts on a state ρAB as follows W = P ′ ′ [ρ⊗ ] (112) n A B − AB ρout = Λ(ρAB)= Ai Bi(ρAB)A† B†, (110) is not an entanglement witness (here A B is two qubit ⊗ i ⊗ i ′ ′ i + X system, P is maximally entangled state). One may also think, that the problem could be solved which after partial transpose on subsystem B gives by use of simpler class of maps - namely PPT operations. Γ T Γ T Remarkably, in (Eggeling et al., 2001) it was shown that ρ = A (B†) (ρ )A† B . (111) out i ⊗ i AB i ⊗ i any NPT state can be distilled by PPT operations. Re- i X cently the problem was attacked by means of positive Γ maps, and associated “witnesses” in (Clarisse, 2005) (see The resulting operator is positive, if only ρAB was posi- tive. also (Clarisse, 2006a)). Since the discovery of first bound entangled states The problem of existence of NPT BE states has impor- quite many further examples of such states were found, tant consequences. If they indeed exist, then distillable only a few of which we have discussed (see Sec. VI.B.7). entanglement is nonadditive and nonconvex. Two states The comprehensive list of achievements in this field, as of zero ED will together give nonzero ED. The set of bi- well as the introduction to the subject can be found in partite BE states will not be closed under tensor product, (Clarisse, 2006b). and under mixing. For the extensive review of the prob- lem of existence of NPT BE states see (Clarisse, 2006b). Schematic representation of the set of all states includ- H. The problem of NPT bound entanglement ing hypothetical set of NPT bound entangled states is shown on fig. 3. Although it is already known that there exist entan- gled nondistillable states, still we do not have a charac- terization of the set of such states. The question which I. Activation of bound entanglement remains open since the discovery of bound entanglement properties of PPT states is: are all NPT states distil- Entanglement is always considered as a resource useful lable? For two main attempts 48 to solve the prob- for certain task. It is clear, that pure entanglement can lem see (DiVincenzo et al., 2000a; D¨ur et al., 2000a). In be useful for many tasks, as it was considered in Sec. III. (Horodecki and Horodecki, 1999) it was shown that this Since discovery of bound entanglement a lot of effort was holds if and only if all NPT Werner states (equivalently devoted to find some nontrivial tasks that this type of entangled Werner states) are distillable. It simply fol- entanglement allows to achieve. lows from the fact, that as in case of two qubits any 49 Equivalently, there exists pure state ψ of Schmidt rank 2 such 48 n Γ | i Two recent attempts are unfortunately incorrect that ψ `ρ⊗ ´ ψ < 0 (DiVincenzo et al., 2000a; D¨ur et al., (Chattopadhyay and Sarkar, 2006; Simon, 2006). 2000a).h | | i 46 2002a), that any NPT state, can be made one-copy dis- tillable51 by use of BE states. Moreover to this end one needs BE states which are arbitrarily close to separable states. nonoptimal A remarkable result showing the power of LOCC op- witness erations supported with arbitrarily small amount of BE Separable states is established in (Ishizaka, 2004). Namely, the in- optimal terconversion of pure bipartite states is ruled by entangle- PPT witness ment measures. In particular, a pure state with smaller measure called Schmidt rank (see Sec. XV) cannot be undistillable NPT? turned by LOCC into a state with higher Schmidt rank. distillable However any transition between pure states is possible with some probability, if assisted by arbitrarily weakly FIG. 3 Schematic representation of the set of all states with bound entangled states. This works also for multipar- example of entanglement witness and its optimization tite states. Interestingly, the fact that one can increase Schmidt rank by PPT operations (i.e. also with some probability by assistance of a PPT state) is implicit al- The first phenomenon which proves usefulness of ready in (Audenaert et al., 2003). bound entanglement, called activation of bound entangle- Although specific examples of activators has been ment, was discovered in (Horodecki et al., 1999a). A pa- found, the general question if any (bound) entangled rameter which is improved after activation is the so called state can be an activator has waited until the discovery (p) probabilistic maximal singlet fraction, that is Fmax(ρ) = of Masanes (Masanes, 2005a,b). He showed, that every max 1 Tr[Λ(ρ) Φ+ Φ+ ] for ρ acting on d d, entangled state (even a bound entangled one) can en- Λ Tr[Λ(ρ)] | ih | C ⊗C where Λ are local filtering operations 50. hance maximal singlet fraction and in turn a fidelity of (p) teleportation of some other entangled state, i.e. for any Consider a state σ with Fmax bounded away from 1, state ρ there exists state σ such that such that no LOCC protocol can go beyond this bound. Then a protocol was found, involving k pairs of some BE (p) (p) Fmax(σ) < Fmax(ρ σ). (114) state ρbe, which takes as an input a state σ, and outputs a ⊗ state σ′ with singlet fraction arbitrarily close to 1 (which This result52 for the first time shows that depends on k). That is: Every entangled state can be used to some nonclas- k (p) • ρ⊗ σ σ′, lim Fmax(σ′)=1 (113) sical task be ⊗ −→ k The protocol is actually closely related to recurrence dis- Masanes provides an existence proof via reductio ad ab- tillation protocol (see Sec. XII.B), generalized to higher surdum. It is then still a challenge to construct for a given dimension and with twirling step removed. In the above state some activator, even though one has a promise that scenario the state ρbe is an activator for a state σ. The it can be found. This result indicates first useful task, probability of success in this protocol decreases as the that can be performed by all bound entangled states. output fidelity increases. To understand the activation The idea of activation for bipartite case was devel- recall that probabilistic maximal singlet fraction of every oped in multipartite case in (D¨ur and Cirac, 2000a) and 1 (Bandyopadhyay et al., 2005) where specific families of bound entangled state is by definition bounded by d , as k multipartite bound entangled states were found. Inter- discussed in Sec. XII.G. It implies, that ρbe⊗ also have (p) estingly, those states were then used to find analogous Fmax bounded away from 1. We have then two states phenomenon in classical key agreement scenario (see Sec. with probabilistic maximal fidelity bounded away from XIX). It was recently generalized by Masanes (again in 1, which however changes if they are put together. For existential way) (Masanes, 2005b). this reason, the effect of activation demonstrates a sort The activation considered above is a result which con- of nonadditivity of maximal singlet fraction. cerns one copy of the state. An analogous result which The effect of activation was further developed in var- hold in asymptotic regime, called superactivation was ious directions. It was shown in (Vollbrecht and Wolf, 51 A state ρ is called one copy distillable, if there exists projectors 50 The superscript (p) emphasizes a ”probabilistic” nature of this P,Q of rank two such that P QρP Q is NPT. In other words, maximal singlet fraction so that it would not be confused from ρ one can then obtain by⊗ LOCC⊗ a twoqubit state with F with different parameter defined analogously as Fmax(ρ) = greater than 1/2. max 1 Tr[Λ(ρ) Φ+ Φ+ ] with Λ being trace preserving 52 Λ Tr[Λ(ρ)] | ih | Actually, the result is even stronger: it holds also for Fmax (i.e. LOCC operation, that is called maximal singlet fraction. singlet fraction achievable with probability one). 47 found in (Shor et al., 2003). Namely there are fourpar- are 2 4 NPT states, but as we already know all NPT tite states, such that no two parties even with help of 2 N⊗states are distillable (see previous section on dis- other parties can distill pure entanglement from them: tillation⊗ of bipartite entanglement). In this way one can distill entanglement between AC and BC which allows to 1 4 distill three particle GHZ entanglement as already previ- ρbe = Ψi Ψi Ψi Ψi . (115) ABCD 4 | ABih AB| ⊗ | CDih CD| ously mentioned. Thus three qubit bound entanglement i=1 X of ̺ABC can be activated with help of two-qubit pure However the following state, consisting of five copies of entanglement. the same state, but each distributed into different parties A simple, important family of bound entangled states is no longer bound entangled are four-qubit Smolin bound entangled states (85). They have the property, that they can be “unlocked”: if parties be be be be be ρfree = ρ ρ ρ ρ ρ (116). AB meet, then the parties CD can obtain an EPR pair. ABCD ⊗ ABCE ⊗ ABDE ⊗ ACDE ⊗ BCDE They are bound entangled, because any cut of the form This result is stronger than activation in two ways. First two qubits versus two other qubits is separable. Again, it turns two totally non-useful states (bound entangled if instead we could distill a pure state, the Smolin state ones) into a useful state (distillable one), and second would be entangled with respect to some cut of this form. the result does not concern one copy but it has asymp- Recently a relationship between multipartite bound totically nonvanishing rate. In other words it shows, entangled states and the stabilizer formalism was founded that there are states ρ and ρ such that E(ρ ρ ) > 1 2 1 ⊗ 2 (Wang and Ying, 2007), which allows to construct a E(ρ1)+ E(ρ2), despite the fact that E(ρ1)= E(ρ2)=0 wide class of unlockable bound entangled states in where E is suitable measure describing effect of distilla- arbitrary multiqudit systems. It includes a gener- tion (see Sec. XV). alized Smolin states (Augusiak and Horodecki, 2006b; Bandyopadhyay et al., 2005). 1. Multipartite bound entanglement In this case one defines bound entanglement as any J. Bell inequalities and bound entanglement entangled state (i.e. violating full separability condition) such that one can not distill any pure entanglement be- Since bound entanglement is a weak resource, it was tween any subset of parties. Following the fact that vi- natural to ask, whether it can violate Bell inequalities. olation of PPT condition is necessary for distillation of The first paper reporting violation of Bell inequalities was entanglement it is immediate that if the state satisfies due to D¨ur (D¨ur, 2001) who showed that some multiqubit PPT condition under any bipartite cut then it is impos- bound entangled states violate two-settings inequalities sible to distill pure entanglement at all. called Mermin-Klyshko inequalities, for n-partite states with n 8. Further improvements (Kaszlikowski et al., The first examples of multipartite bound entanglement ≥ were semiseparable three qubit states ̺Shift (see Sec. 2000) (n 7) (Nagata et al., 2006; Sen(De) et al., 2002) ABC (n 6) have≥ been found. In (Augusiak et al., 2006) the VII.B). Immediately it is not distillable since it is sepa- ≥ rable under any bipartite cut. Thus not only pure entan- lowest so far number of particles was established: it was glement, but simply no bipartite entanglement can be shown that the four-qubit Smolin bound entangled states obtained between any two subsystems, while any pure (85) violate Bell inequality. The latter is very simple multipartite state possesses entanglement with respect distributed version of CHSH inequality: to some cut. a a a (b +b′ )+a′ a′ a′ (b b′ ) 2 One can relax the condition about separability or PPT | A ⊗ B ⊗ C ⊗ D D A ⊗ B ⊗ C ⊗ D − D |≤ property and the nice example is provided in (D¨ur et al., (117) We see that this is the usual CHSH inequality where 1999b). This is an example of 3-qubit state ̺ABC from the family (86) such that it is PPT with respect to one party was split into three, and the other on (B) was two partitions but not with respect to the third one i.e. untouched. What is curious, the maximal violation is TA TB TC obtained exactly at the same choices of observables, as ̺ABC 0, ̺ABC 0 but ̺ABC 0. Thus not all non- trivial≥ partitions have≥ NPT property6≥ which, in case of for singlet state with standard CHSH: this family, was proven to be equivalent to distillability (see last paragraph of the previous subsection). Still the a = σx, a′ = σy, state is entangled because of violation PPT along the σx + σy σx σy b = , b′ = − (118) AB C partition, hence it is bound entangled. √2 √2 Note,| however, that the bound entanglement in this state ̺ABC can be activated with partial free entangle- and the value is also the same as for singlet state: 2√2. + ment of the state ΦAB 0 C since then part A of bound Werner and Wolf analyzed multipartite states (Werner, entangled state can| bei| teleportedi to B and vice versa, 2001b; Werner and Wolf, 2000) and have shown that producing in this way two states σAC = σBC between Al- states which are PPT in every cut (we will call them ice and Charlie and Bob and Charlie respectively which PPT states in short) must satisfy any Bell inequality of 48 type N 2 2 (i.e. involving N observers, two observables each other. Thus generically, LOCC transformations be- per site,× each× observable with two outcomes). tween pure states are irreversible. However as we will The relation of Bell inequalities with bipartite distilla- see, it can be lifted in asymptotic limit. bility has been also analyzed in (Acin et al., 2006b) where Further important results in this area have been pro- it has been shown that any violation of Bell inequalities vided by (Vidal, 2003) who obtained optimal probability from the Mermin-Klyshko class always leads to possibil- of success for transitions between pure states (see Sec. ity of bipartite entanglement distillation between some XV.D.1), and (Jonathan and Plenio, 1999), who consid- subgroups of particles. ered transitions state ensemble. In 1999 Peres has conjectured (Peres, 1999) that PPT → states do not violate Bell inequalities: 1. Entanglement catalysis LHVM PPT. (119) ⇔ The most surprising consequence of the Nielsen’s We see that the above results support this conjecture. laws of pure state transitions have been discovered in The answer to this question, if positive, will give very im- (Jonathan and Plenio, 1999). Namely, for some states portant insight into our understanding of classical versus ψ1 and ψ2 for which transition ψ1 ψ2 is impossible, quantum behavior of states of composite systems. the following process is possible: → Let us finally mention, that all those considerations concern violation without postselection, and operations ψ1 φ ψ2 φ, (121) applied to many copies. Recently, Masanes showed, that ⊗ → ⊗ if we allow collective manipulations, and postselection, thus we borrow state φ, run the transition, and obtain no bound entangled states violate the CHSH inequality untouched φ back! The latter state it plays exactly a role even in asymptotic regime (Masanes, 2006). of catalyst: though not used up in the reaction, its pres- ence is necessary to run it. Interestingly, it is not hard to see that the catalyst cannot be maximally entangled. XIII. MANIPULATIONS OF ENTANGLEMENT AND The catalysis effect was extended to the case of mixed IRREVERSIBILITY states in (Eisert and Wilkens, 2000). A. LOCC manipulations on pure entangled states — exact case 2. SLOCC classification The study on exact transformations between pure For multipartite pure states there does not exist states by LOCC was initiated by (Lo and Popescu, Schmidt decomposition. Therefore Nielsen result cannot 2001). A seminal result in this area is due to Nielsen be easily generalized. Moreover analysis of LOCC manip- (Nielsen, 1999). It turns out that the possible tran- ulations does not allow to classify states into some coarse sitions can be classified in a beautiful way in terms grained classes, that would give a rough, but more trans- of squares of Schmidt coefficients λ (i.e. eigenval- i parent picture. Indeed, two pure states can be trans- ues of local density matrix). Namely, a pure state formed into each other by LOCC if and only if they can d (ψ) ψ = j=1 λj jj can be transformed into other be transformed by local unitary transformations so that | i | i to parameterize classes one needs continuous labels, even pure stateP φ q= d λ(φ) jj if and only if for each | i j=1 j | i in bipartite case. k 1,...,d there holdsq To obtain a simpler, “coarse grained” classification, ∈{ } P which would be helpful to grasp important qualitative k k (ψ) (φ) features of entanglement, it was proposed (D¨ur et al., λ ↓ λ ↓, (120) j ≤ j 2000b) to treat states as equivalent, if with some nonzero j=1 j=1 X X probability they can be transformed into each other by LOCC. This is called stochastic LOCC, and denoted by (ψ,φ) where λj ↓ are eigenvalues of subsystem of ψ (φ) in SLOCC. It is equivalent to say that there exists reversible descending order. The above condition states that φ ma- operators Ai such that jorizes ψ (see also Sec. (V.C)). Thus one can transform ψ into φ only when subsystems of ψ are more mixed ψ = A1 . . . AN φ . (122) than those of φ. This is compatible with Schr¨odinger ap- | i ⊗ ⊗ | i proach: the more mixed subsystem, the more entangled For bipartite pure states of d d system we obtain in this state. way d entangled classes of states,⊗ determined by number Since majorization constitutes a partial order, re- of nonzero Schmidt coefficients (so called Schmidt rank). versible conversion ψ φ is possible if and only if the Here is example of SLOCC equivalence: the state Schmidt coefficients of↔ both states are equal. Moreover there exist states either of which cannot be converted into ψ = a 00 + b 11 (123) | i | i | i 49 with a>b> 0 can be converted (up to irrelevant phase) (1/3, 1/3, 1/3), but in different class than (0.5, 0.5, 0), b 0 while we clearly see that the first and the last have into φ+ by filter A I, with A = a with probabil- | i ⊗ 0 1 much more in common than the middle one. In order to ity p = 2 b 2; so it is possible to consider representative neglect small differences, one can employ some asymp- state for each| | class. totic limit. This is in spirit of Shannon’s communica- A surprising result is due to Ishizaka (Ishizaka, 2004) tion theory, where one allows for some inaccuracies of in- who considered SLOCC assisted by bound entangled formation transmission, provided they vanish in asymp- PPT states. He then showed that every state can be con- totic limit of many uses of channel. Interestingly, the verted into any other. This works for both bipartite and first results on quantitative approach to entanglement multipartite pure states. For multipartite states SLOCC (Bennett et al., 1996b,c,d) were based on LOCC trans- classification was done in the case of three (D¨ur et al., formations in asymptotic limit. 2000b) and four qubits (Verstraete et al., 2002), and also In asymptotic manipulations, the main question is two qubits and a qudit (Miyake and Verstraete, 2004). what is the rate of transition between two states ρ and For three qubits there are five classes plus fully product σ. One defines the rate as follows. We assume that Alice state, three of them being Bell states between two qubits and Bob have initially n copies in state ρ. They apply (i.e. states of type EPR 0 ). Two others are GHZ LOCC operations, and obtain m pairs53 in some joint AB ⊗ | iC state state σm. If for large n the latter state approaches state σ m, i.e. 1 ⊗ GHZ = ( 000 + 111 ) (124) m | i √2 | i | i σ σ⊗ 0 (126) k m − k1 → and so called W state and the ratio m/n does not vanish, then we say that ρ 1 can be transformed into σ with rate R = lim m/n. The W = ( 100 + 010 + 001 ). (125) | i √3 | i | i | i largest rate of transition we denote by R(ρ σ). In particular, distillation of entanglement described→ in Sec. They are inequivalent, in a sense, that none of them can XII is the rate of transition to EPR state be converted into the other one with nonzero probability + (unlike in bipartite state case, where one can go from any ED(ρ)= R(ρ ψ ). (127) class to any lower class i.e. having lower Schmidt rank). → In 2 2 d case (Miyake, 2004; Miyake and Verstraete, The cost of creating state out of EPR states is given by 2004),⊗ there⊗ is still discrete family of inequivalent + classes, where there is maximally entangled state — two EC (ρ)=1/R(ψ ρ) (128) + + → EPR state φAB1 φB2C (where the system B is four- dimensional). Any⊗ state can be produced from it simply and it is the other basic important measure (see Sec. via teleportation (Bob prepares the needed state, and XV.A for description of those measures in more detail). teleports its parts to Alice and to Charlie. In four qubit case the situation is not so simple: the inequivalent classes constitute a continuous family, which 1. Unit of bipartite entanglement one can divide into nine qualitatively different subfami- lies. The fundamental result in asymptotic regime is that The SLOCC classification is quite elegant generaliza- any bipartite pure state can be transformed into two- tion of local unitary classification. In the latter case the qubit singlet with rate given by entropy of entanglement basic role is played by invariants of group SUd1 . . . SA = SB , i.e. entropy of subsystem (either A or B, since SU for d . . . d system, while in SLOCC,⊗ the⊗ for pure states they are equal). And vice versa, to create dN 1 ⊗ ⊗ N relevant group is SLd1, . . . SLdN , (one restricts any state from two-qubit singlet, one needs SA singlets to filters of determinantC 1,⊗ because⊗ the normalizationC of pair pair two-qubit state. Thus any pure bipartite state states does not play a role in SLOCC approach). can be reversibly transformed into any other state. As a Finally, the SLOCC classification of pure states can be result, in asymptotic limit, entanglement of these states used to obtain some classification of mixed states (see can be described by a single parameter — von Neumann (Acin et al., 2001; Miyake and Verstraete, 2004)). entropy of subsystem. Many transitions that are not al- lowed in exact regime, become possible in asymptotic limit. Thus the irreversibility implied by Nielsen result B. Asymptotic entanglement manipulations and is lifted in this regime, and EPR state becomes universal irreversibility unit of entanglement. The classifications based on exact transformations suffer for some lack of continuity: for example in SLOCC approach ψ with squares of Schmidt coeffi- 53 Here m depends on n, which we do note write explicitly for cients (0.5, 0.49, 0.01) is in the same class as the state brevity. 50 2. Bound entanglement and irreversibility for trivial reasons. Example of such state is 1 + + 1 However, even in asymptotic limit one cannot get ψ ψ 0 A′ 0 + ψ− ψ− 1 A′ 1 . (129) 2| ABih AB |⊗| i h | 2| ABih AB |⊗| i h | rid of irreversibility for bipartite states, due to exis- tence of bound entangled states. Namely, to create such The states 0 , 1 are local orthogonal “flags”, which | i | i a state from pure states by LOCC one needs entan- allow to return to the pure state ψ+ on system AB. gled states, while no pure entanglement can be drawn This shows that, within mixtures of maximally entan- back from it. Thus the bound entangled state can be gled states, irreversibility is a generic phenomenon. Last viewed as a sort of black holes of entanglement theory but not least, it was shown that irreversibility is exhib- (Terhal et al., 2003b). One can also use thermodynam- ited by all bound entangled states (Yang et al., 2005a) ical analogy (Horodecki et al., 2002, 1998b). Namely, (see Sec. XIII). bound entangled state is like a single heat bath: to cre- One could still try to regain reversibility in some form. ate the heat bath, one needs to dissipate energy, but Recall the mentioned thermodynamical system with dif- no energy useful to perform mechanical work can be ference of temperatures. Since only part of the en- drawn in cyclic process (counterpart of the work is here ergy can perform useful work, we have irreversibility. quantum communication via teleportation). Note in this However the irreversible process occurred in the past, context, that the interrelations between entanglement while creating the system by partial dissipation of en- and energy were considered also in different contexts ergy. However, once the system is already created, the (see e.g. (Horodecki et al., 2001c; McHugh et al., 2006; work can be reversibly drawn and put back to the system Osborne and Nielsen, 2002)). in Carnot cycle. In the case of entanglement, it would One might hope to regain reversibility as follows: per- mean that entanglement can be divided into two parts: haps for many copies of bound entangled state ρ, though bound entanglement and free (pure entanglement) which some pure entanglement is needed, a sublinear amount then can be reversibly mixed with each other. One can test such hypotheses by checking first, whether having would be enough to create them (i.e. the rate EC van- ishes). In other words, it might be that for bound entan- bound entanglement for free, one can regain reversibil- ity (Horodecki et al., 1998a). In (Horodecki et al., 2002) gled states EC vanishes. This would mean, that asymp- totically, the irreversibility is lifted. strong evidence was provided, that even this is not the case, while conversely in (Audenaert et al., 2003) it was More general question is whether distillable entangle- shown, that to some extent such reversibility can be ment is equal to entanglement cost for mixed states. Al- reached (see also (Ishizaka, 2004) in this context) 55. ready in the original papers on entanglement distilla- tion (Bennett et al., 1996c,d) there was indication, that generically we would have a gap between those quanti- 3. Asymptotic transition rates in multipartite states ties54, even though for some trivial cases we can have EC = ED for mixed states ((Horodecki et al., 1998b)). In multipartite case there is no such a universal unit of Continuing thermodynamical analogy, a generic mixed entanglement as the singlet state. Even for three parti- state would be like a system of two heat baths of differ- cles, we can have three different types of EPR states: the ent temperature, from which part of energy but not the one shared by Alice and Bob, by Alice and Charlie and whole can be transferred into mechanical work. Bob and Charlie. By LOCC it is impossible to create It was a formidable task to determine, if we have irre- any of them from the others. Clearly, one cannot create + + versibility in asymptotic setting, as it was related to fun- at all φ AB from φ BC because the latter has zero en- 56 + damental, and still unsolved problem of whether entan- tanglement across A : BC cut . From two states φ AB + + glement cost is equal to entanglement of formation (see and φ AC one can create φ BC via entanglement swap- Sec. XV). The first example of states with asymptotic ping, i.e. by teleportation of half of one of pairs through irreversibility was provided in (Vidal and Cirac, 2001). the other pair. However this is irreversible: the obtained + Subsequently more and more examples have been re- state φ BC does not have entanglement across A : BC + vealed. In (Vollbrecht et al., 2004) mixtures of maxi- cut, so that it is impossible to create state φ AB and for + mally entangled states were analyzed by use of uncer- similar reason also the state φ AC . tainty principle. It turns out that irreversibility for this To see why even collective operations on many copies class of states is generic: the reversible states happen + n + m + k cannot make the transformation φ AB⊗ + φ AC⊗ φ AB⊗ to be those which minimize uncertainty principle, and reversible it is enough to examine entropies of→ subsys- all they turn out to be so called pseudo-pure states (see tems. For pure states they do not increase under LOCC, (Horodecki et al., 1998b)), for which reversibility holds 55 It should be noted however, that in (Audenaert et al., 2003) PPT 54 Although in their operational approach the authors meant what operations have been used, which may be a stronger resource, we now call entanglement cost, to quantify it they used non- than assistance of PPT states. 56 + + + + regularized measure entanglement of formation. By φ we mean φ 0 C , similarly for φ and φ . AB AB ⊗| i AC BC 51 so they have to be constant during reversible process. XIV. ENTANGLEMENT AND QUANTUM in out Thus we have to have : SX = SX for X = A, B, C COMMUNICATION which gives In classical communication theory, the most important notion is that of correlations. To send a message, means A : n + m = k; B : n = k; C : m = k. (130) in fact to correlate sender and receiver. Also the famous Shannon formula for channel capacity involves mutual Thus we arrived at contradiction (to be rigorous, we information, a function describing correlations. Thus should apply here asymptotic continuity of entropy, see the ability to faithfully transmit a bit is equivalent to Sec. XV.B.3). In the described case irreversibility clearly the ability to faithfully share maximally correlated bits. enters, however the reason for this is not deep. It was early recognized that in quantum communication theory it is entanglement which will play the role of cor- Via such considerations, one can see that in four- relations. In this way entanglement is the cornerstone of particle case not only EPR states are “independent” quantum communication theory. units, but also GHZ state cannot be created from them, In classical communication theory, a central task is to hence it constitutes another independent unit. For three send some signals. For a fixed distribution of signals particles it is less immediate, but still true that GHZ can- emitted by a source, there is only one ensemble of mes- not be created reversibly from EPR states (Linden et al., sages. In quantum case, a source is represented by a 1999a). This shows that even in asymptotic setting density matrix ρ, and there are many ensembles realizing there is truly tripartite entanglement, a distinct quality the same density matrix. What does it then mean to send that cannot be reduced to bipartite entanglements. This quantum information? According to (Bennett et al., shows that there is true three-particle entanglement in 1993) it is the ability of transmitting and unknown quan- asymptotic limit. tum state. For a fixed source, this would mean, that all + Irreversibility between GHZ and φ (much less triv- possible ensembles are properly transmitted. For exam- ial than the one between| EPR statesi themselves) can be 1 ple, consider a density matrix ρ = d i i i . Suppose also seen as consequence of the fact that all two-particle that a channel decoheres its input in basis| ih | i . We reduced density matrices of GHZ are separable, while see that the set of states i goes throughP {| thei} chan- | i for EPR states two are separable and one is entangled. In nel without any disturbance.{| i} However complementary bipartite states we cannot transform separable state into set consisting of states U i where U is discrete Fourier entangled one by LOCC at all. Here, the transition is transform, is completely destroyed| i by a channel, because possible, because the third party can help, however this the channel destroys superpositions. (For d = 2, exam- requires some measurement which introduces irreversibil- ple of such complementary ensemble is + , where ity. = 1 ( 0 1 )). As a matter of fact,| eachi |−i member |±i 2 | i ± | i Irreversibility was further explored in (Acin et al., of the complementary ensemble is turned into maximally 2003d) and irreversibility on the level of mixed bipar- mixed state. tite states was used to show that even GHZ plus EPR’s How to recognize whether all ensembles can go do not constitute what is called minimal reversible en- through? Schumacher has noted (Schumacher, 1995a) tanglement generating set (Bennett et al., 2001) i.e. a that instead of checking all ensembles we can check minimal set of states from which any other state can be whether an entangled state ψAB is preserved, if we reversibly obtained by LOCC. Such set is still unknown send half of it (the system B) down the channel. The even in the case of pure states of three qubits. Recently state should be chosen to be purification of ρ, i.e. it was shown (Ishizaka and Plenio, 2005) that for three TrA( ψ ψ AB ) = ρ. Thus sending an unknown state | ih | qubits the set cannot be also constituted by EPR’s and is equivalent to sending faithfully entanglement. 1 Indeed, in our example, the state can be chosen as W state W = √ ( 001 + 010 + 100 ). In (Vidal et al., 3 | i | i | i Φ+ = 1 i i . One can see that after applying 2000) a nontrivial class of states was shown, which are | i √d i | i| i our channel to one subsystem, the state becomes classi- reversibly transformable into EPR’s and GHZ: P cal (incoherent) mixture of states i i . This shows that the channel cannot convey quantum| i| informationi at all. 1 ψ ABC = c0 000 + c1 1 ( 11 + 22 ). (131) It is reflection of a mathematical fact, that if we send | i | i | i√2 | i | i half of purification of a full rank density matrix down the channel, then the resulting state will encode all the parameters of the channel. This heuristic statement has The irreversibility between pure multipartite states its mathematical form in terms of Choi-Jamio lkowski iso- seems to be a bit different than that of mixed states in morphism between states and channels. Its most stan- that it is less “thermodynamical”. In the former case we Λ dard form links the channel Λ with a state ̺AB having have two “good” states, which cannot be transformed re- Λ 1 maximally mixed left subsystem TrB(̺ )= I as fol- versibly into one another (both EPR state and GHZ state AB dA lows: are useful for some tasks), while bound entanglement is Λ + + clearly “worse” than free entanglement. ̺ = [I Λ ] Φ ′ Φ ′ , (132) AB A ⊗ B | AA ih AA | 52 + Λ where the projector onto maximally entangled state PAA′ which is equal to overlap of the state ̺ with max- Λ is defined on a product Hilbert space A A′ , with imally entangled state namely F (Λ) = F (̺ ) := ′ 57 H ⊗ H + + Λ A A . Tr( Φd Φd ̺ ). It is interesting that one has H ≃ H (Horodecki| ih et| al., 1999b; Nielsen, 2002): dF (Λ)+1 f(Λ) = . (136) A. Capacity of quantum channel and entanglement d +1 This formula says that possibility of sending on average The idea of quantum capacity Q(Λ) of quantum chan- faithfully quantum information happens if and only if it nel Λ was introduced in the seminal work (Bennett et al., is possible to create maximal entanglement of Φ+ with 1996d) which contains milestone achievements connect- d help of the channel. The above relation is an important ing quantum entanglement and quantum data transfer. element in the proof that, quite remarkably, definition of The capacity Q measures the largest rate of quantum in- formation sent asymptotically faithfully down the chan- zero-way (or — alternatively — one-way forward) version of quantum capacity Q (see below) remains the same if nel: we apply any of the fidelities recalled above (for details # transmitted faithfully qubits of the proof see the review (Kretschmann and Werner, Q = sup , (133) # uses of channel 2004)). Even more interesting, LHS of the above equality can where the fidelity of the transmission is measured by be interpreted as an average teleportation fidelity of the minimal subspace fidelity f(Λ) = min ψ Λ( ψ ψ ) ψ channel that results from teleporting given state through ψh | | ih | | i (Bennett et al., 1997). Also average fidelity transmis- the mixed bipartite state ̺Λ. sion can be used which is direct analog of average An impressive connection between entanglement and fidelity in quantum teleportation process: f(Λ) = quantum channels theory has been worked out already in ψ Λ( ψ ψ ) ψ dψ with uniform measure dψ on unit ((Bennett et al., 1996c,d)) by use of teleportation. The sphere.h | | ih | | i authors have shown how to achieve nonzero transmis- R Other approach based on idea described above was for- sion rate by combining three elements (a) creating many Λ malized in terms of entanglement transmission. In par- copies of ̺ by sending halves of singlets down the chan- ticular, the quality of transmission was quantified by en- nel Λ (b) distilling maximal entanglement form many tanglement fidelity copies of the created state (c) teleporting quantum in- formation down the (distilled) maximal entanglement. F (Λ, ΨAB)= ΨAB [IA ΛB] ΨAB ΨAB ) ΨAB , Since the last process corresponds to ideal transmission h | ⊗ | ih | | (134)i the rate of the quantum information transmission is here 58 with respect to a given state ΨAB . The alterna- equal to distillation rate in step (ii). In this way one tive definition of quantum capacity (which has been can prove the inequality linking entanglement distillation worked out in (Barnum et al., 1998; Schumacher, 1996; ED with quantum channel capacity Q as (Bennett et al., Schumacher and Nielsen, 1996) and shown to be equiva- 1996d): Λ lent to the original one of (Bennett et al., 1996d, 1997) in ED(̺ ) Q(Λ), (137) (Barnum et al., 2000)) was based on counting the opti- ≤ Λ mal pure entanglement transmission under the condition where ̺ is given by (132). The inequality holds for of high entanglement fidelity defined above. one-way forward an two-way scenarios of distillation (re- A variation of the entanglement fidelity spectively — coding). The above inequality is one of the (Reimpell and Werner, 2005) is when input is equal to central links between quantum channels and quantum en- tanglement theory (see discussion below). It is not known the output dA = dB = d and we send half of maximally Λ + whether there is lower bound like cQ(Λ) ED(̺ ) for entangled state Φd down the channel. It is measured by ≤ maximal entanglement fidelity of the channel: some constant c. However there is at least qualitative equivalence shown in (Horodecki, 2003e): F (Λ) := F (Λ, Φ+), (135) E (̺Λ)=0 Q(Λ) = 0. (138) d D ⇒ An alternative simple proof which works also for multipartite generalization of this problem can be 57 We note that this isomorphism has operational meaning, in gen- found in (D¨ur et al., 2004). It uses teleportation and eral, only one way: given the channel, Alice and Bob can obtain Choi-Jamio lkowski isomorphism to collective channels bipartite state, but usually not vice versa. However sometimes (Cirac et al., 2001). if Alice and Bob share mixed bipartite state, then by use of clas- sical communication they can simulate the channel. Example is maximally entangled state, which allows to regain the corre- B. Fidelity of teleportation via mixed states sponding channel via teleportation. This was first pointed out and extensively used in (Bennett et al., 1996d). 58 For typical source all three types of fidelity — average, minimum Before entanglement distillation was discovered, the and entanglement one are equivalent (Barnum et al., 2000). fundamental idea of teleporting quantum states with help 53 of mixed states has been put forward in the pioneering state we immediately see from (138) that the capacity of paper (Popescu, 1994). The fidelity of transmission that entanglement breaking channel is zero i.e. no quantum was used there was faithful quantum transmission is possible with entangle- ment breaking channel. However the converse is not true: f (̺ )= dφ φ ̺φ φ , (139) possibility to create entanglement with help of the chan- tel AB h | B | i Z nel is not equivalent to quantum communication and it is bound entanglement phenomenon which is responsible with ̺φ being the output state of the teleportation B for that. To see it let us observe that if the corresponding in Bobs hands, i.e. it is average fidelity mentioned state ̺Λ is bound entangled, then the channel Λ, called above. Popescu used this formula to show that tele- in this case binding entanglement channel (introduced portation through some entangled 2-qubit Werner states in (DiVincenzo et al., 2003a; Horodecki et al., 2000d)), that satisfy some local hidden variable models (Werner, clearly allows for creation of entanglement. However it 1989a) beats the classical “teleportation” fidelity thresh- cannot convey quantum information at all — it has ca- old 2/3. Further the formula for f was optimized in the tel pacity zero. This was first argued in (Horodecki et al., case of pure (Gisin, 1996a) and mixed (Horodecki et al., 2000d) and proof was completed in (Horodecki, 2003e) 1996b) 2-qubit states ρ , with Alice’s measurement re- AB just via implication (138)). stricted to maximally entangled projections. For pure Note that binding entanglement channels are not com- state ρ general optimization was performed in d d AB pletely useless from general communication point of view, case and it was shown to be attained just on maxi-⊗ for they can be used to generate secure cryptographic key, mally entangled Alice’s projections (Banaszek, 2000)(see see Sec. (XIX). (Bowen and Bose, 2001) for further analysis). All that teleportation schemes that we have considered so far were deterministic. The idea of probabilistic teleportation was D. Quantum Shannon theorem introduced in (Brassard et al., 2004; Mor, 1996). It was employed in effect of activation of bound entanglement How many qubits can be sent per use of a given chan- (see Sec. XII.I). nel? It turns out that the answer to the question is on Finally let us note that all possible exact teleporta- one hand analogous to classical Shannon formula, and tion and dense coding schemes were provided in (Werner, on the other hand entirely different. Capacity of classi- 2001a). There it was proved, that they are essentially cal channel is given by maximum of mutual information of the same kind. In teleportation of a state of a d- over all bipartite distributions, that can be obtained by dimensional system, Alice measures in a maximally en- use of channel. tangled basis59 ψ . Subsequently Bob decodes by use { i} Quantumly, even for sending qubits, as already men- of unitary transformations related to the basis via Ui + ⊗ tioned above, we may have different scenarios: 1) quan- I Ψ = ψi (similarly as in dense coding). Note that for tum channel without help of classical channel (capacity two| qubitsi | alli maximally entangled bases are related by denoted by Q∅) 2) with help of classical channel: one- local unitary transformation (Horodecki and Horodecki, way forward (Q→) , backward (Q←), and two-way (Q↔). 1996). Thus there is in a sense a unique protocol of tele- There is elegant answer in the case 1), which is called porting a qubit. However in higher dimension it is no Quantum Shannon Theorem. The first proof, partially longer the case (W´ojcik et al., 2003) which leads to in- heuristic, was provided in (Lloyd, 1997), subsequently it equivalent teleportation/dense coding protocols. was improved in (Shor, 2002a) and finally first fully rigor- ous result has been obtained in (Devetak, 2003). Namely, to get asymptotic capacity formula for Q one maximizes C. Entanglement breaking and entanglement binding ∅ coherent information over all bipartite states resulting channels from a pure state, half of it sent down the channel (see Sec.XII.F). However, one should optimize this quantity The formulas (137) and (138) naturally provoke the over many uses of channel, so that the formula for capac- question: which channels have capacity zero?. ity reads Now, if the state ̺Λ is separable, then the cor- responding channel is called entanglement breaking 1 coh n Q∅(Λ) = lim sup IA B((I Λ⊗ ) ψ ψ ). (140) (Horodecki et al., 2003d) and one cannot create entan- n ψ i ⊗ | ih | glement at all by means of such channel.60 Now since it is impossible to distill entanglement from separable The fact that the formula is not “single letter”, in the sense that it involves many uses of channel 59 That is performs a measurement with Kraus operators being d d the projects onto bipartite states ψi which have that action of any channel of that kind can be simulated by maximally mixed subsystems and form| i a ∈ basis C ⊗ of C d d classical channel: for any input state, one measures it via some 60 Simply, its Kraus operators (in some decomposition)C ⊗ are C of rank POVM, and sends the classical outcome to the receiver. Based one. Impossibility of creating entanglement follows from the fact on this, receiver prepares some state. 54 was established in (Shor and Smolin, 1996), see also where σz is a Pauli matrix. Applying the isomorphism (DiVincenzo et al., 1998b). (132) we get rank-two Bell diagonal state Coherent information can be nonzero only for entan- Λp + + ̺ = p φ φ + (1 p) φ− φ− , (143) gled states. Indeed, only entangled states can have | ih | − | ih | greater entropy of subsystem than that of total system 1 with φ± = ( 0 0 1 1 ). The formula for distil- (Horodecki and Horodecki, 1994). In this context, an | i √2 | i| i ± | i| i additional interesting qualitative link between entangle- lable entanglement has been found in this case (Rains, ment and channel capacity formula is the hashing in- 1999, 2001) 61 namely coh equality stating that E→ I (see Sec. XII). Quite D A B Λp coh Λp ≥ i coh ED→(̺ )= IB A(̺ )=1 H(p), (144) remarkably sometimes ED→(̺AB) > 0 for IA B(̺AB)=0 i − i which is related to the above mentioned Shor-Smolin re- where H(p)= p log p (1 p) log(1 p). (Actually, for sult. − 62− − − this state ED = ED→). Finally, there is a remarkable result stating that for- Since it is Bell diagonal state we know that capacity ward classical communication does not increase quantum Q→ can be achieved by protocol consisting of one-way capacity distillation of entanglement from ̺Λp followed by tele- portation (see discussion above). Therefore Q→(Λp) = Q∅ = Q→. (141) coh Λp Q∅(Λp)= IB A(̺ )=1 H(p). i − It was argued in (Bennett et al., 1996d) and the proof was completed in (Barnum et al., 2000). Therefore, any F. Other capacities of quantum channels one-way distillation scheme, provides (through relation with Q→) lower bound for capacity Q∅. One might ask weather maximizing quantum mutual information for a given channel (in analogy to maximiz- ing coherent information, Sec. XIV.D) makes sense. Is E. Bell diagonal states and related channels such a maximized quantity linked to any capacity quan- tity? Recall that quantum mutual information is defined An intriguing relation between states and channels was (in analogy to its classical version) as follows: found (Bennett et al., 1996d) — namely there is a class I (̺ )= S(̺ )+ S(̺ ) S(̺ ). (145) of channels Λ for which there is equality in (137) be- A:B AB A B − AB cause of the following equivalence: not only given Λ Λ First of all one can prove that the corresponding one can produce ̺ (which is easy — one sends half (maximized) quantity would be nonzero even for en- of maximally entangled state Φ+ own the channel) but Λ tanglement breaking channels (Adami and Cerf, 1997; also given ̺ and possibility of additional forward (i.e. Belavkin and Ohya, 2001). Thus it cannot describe from sender to receiver) classical communication one can quantum capacity (since, as we already know, they have simulate again the channel Λ. There is a fundamen- quantum capacity zero). Moreover, for a qubit channel it tal observation (Bennett et al., 1996d)(which we recall can be greater than 1, for example for noiseless channel it here for qubit case but d d generalization is straight- ⊗ is just 2. This reminds dense coding, where two bits are forward, see (Horodecki et al., 1999b)) that if the state send per sent qubit, with additional use of shared EPR ̺Λ is Bell diagonal than the operation of teleporting any Λ pair. And these two things are indeed connected. It state σ through mixed state ̺ produces just Λ(σ) in was established in (Bennett et al., 1999c, 2002) that the Bobs hands. Thus in this special case given state iso- maximum of quantum mutual information taken over the morphic to the channel plus possibility of forward classi- states resulting from channel is exactly the classical ca- cal communication one can simulate action of the chan- pacity of a channel supported by additional EPR pairs (it nel. Thus for Bell diagonal states, we have an operational is so called entanglement assisted capacity). Moreover, isomorphism, rather than just mathematical one (132). it is equal to twice quantum capacity: In (Bennett et al., 1996d) this was exploited to show Λ that there is equality in (137) i.e. ED→(̺ ) = Q→(Λ), C =2Q = sup I (I Λ )( ψ ψ ) (146) Λ ass ass A:B A B AB and ED(̺ ) = Q↔(Λ). The protocol attaining capac- ⊗ | i h | ity of the channel is in this case sending half of ψ− = where supremum is taken over all pure bipartite states. 1 | i ( 01 10 ) and perform the optimal entanglement Note that the expression for capacity is “single letter”, √2 | i − | i distillation protocol. To summarize the above description consider now the example where this analysis finds application. 61 That 1 H(p) is achievable was shown in (Bennett et al., 1996d). Example .- Consider the quantum binary Pauli chan- 62 − nel acting on single qubit as follows: Note that this is equal to capacity of classical binary symmetric channel with error probability p. However, the classical capacity of the present channel is maximal, because classical information Λ (ρ)= p(ρ)+(1 p)σ (ρ)σ , (142) can be sent without disturbance through phase. p − z z 55 i.e. unlike in the unassisted quantum capacity case, it is question of additivity of classical capacity C(Λ), a possi- enough to optimize over one use of channel. ble role of entanglement in inputs of the channel Λ1 Λ2 A dual picture is, when one has noiseless channel, sup- comes into play. ⊗ ported by noisy entangled states. Again, the formula in- Though this problem is still open, there is its multipar- volves coherent information (the positive coherent infor- tite version where nonadditivity was found (D¨ur et al., mation says that we have better capacity, than without 2004). In fact, the multiparty communication scenario support of entanglement) (Bowen, 2001; Horodecki et al., can be formulated and the analog of (137) and (138) can 2001b). There are also interesting developments con- be proved (D¨ur et al., 2004) together with binding entan- cerning multipartite dense coding (Bruß et al., 2004; glement channels notion and construction. Remarkably Horodecki and Piani, 2006). in this case the application of multipartite BE channels isomorphic to multipartite bound entangled states and application of multipartite activation effect leads to non- G. Additivity questions additivity of two-way capacity regions for quantum broad- cast (equivalently multiply access) channel (D¨ur et al., There is one type of capacity, where we do not send half 2004): tensor product of three binding channels Λi which of entangled state, but restrict just to separable states. automatically have zero capacity leads to the channel This is classical capacity of quantum channel C(Λ), with- 3 Λ : i=1Λi with nonzero capacity. In this case bound out any further support. However even here entangle- entanglement⊗ activation (see section on activation) leads ment comes in. Namely, it is still not resolved, whether to the proof of nonadditivity effect in quantum informa- sending signals entangled between distinct uses of chan- tion transmission. Existence of the alternative effect for nel can increase transmission rate. This problem is equiv- one-way or zero-way capacity is an open problem. Finally alent to the following one: can we decrease production of let us note in this context that the multipartite general- entropy by an operation Λ, by applying entangled inputs, ization of the seminal result Q∅ = Q→ has been proved to Λ Λ? i.e. ⊗ (Demianowicz and Horodecki, 2006) exploiting the no- tion of entanglement transmission. 2 inf S(Λ(ρ)) =? inf S(Λ Λ(ρ)). (147) ρ ρ ⊗ There is an interesting application of multipartite bound entanglement in quantum communication. As we One easily finds that if this equality does not hold have already mentioned, bipartite bound entanglement, it may happen only via entangled states. Quite in- which is much harder to use than multipartite one, is able terestingly, these problems have even further connec- to help in quantum information transfer via activation tion with entanglement: they are equivalent to addi- effect (Horodecki et al., 1999a). There is a very nice ap- tivity of one of the most important measures of en- plication of multipartite bound entanglement in remote tanglement — so called entanglement of formation, see quantum information concentration (Murao and Vedral, (Audenaert and Braunstein, 2004; Koashi and Winter, 2001). This works as follows: consider the 3-qubit state 2004; Matsumoto, 2005; Shor, 2003) and references ψABC (φ) being an output of quantum cloning machine therein. We have the following equivalent statements that is shared by three parties Alice, Bob and Charlie. additivity of entanglement of formation The initial quantum information about cloned qubit φ • has been delocalized and they cannot concentrate it back superadditivity of entanglement of formation in this distant lab scenario. If however each of them • is given in addition one particle of the 4-particles in a additivity of minimum output entropy unlock • Smolin state ̺ABCD (85) with remaining fourth D parti- additivity of Holevo capacity cle handed to another party (David) then a simple LOCC • action of the three parties can “concentrate” the state φ additivity of Henderson-Vedral classical correlation back remotely at David’s site. • measure CHV (see Secs. XV and XIII for definitions of EF and CHV ). To prove or disprove them accounts for one of the funda- H. Miscellanea mental problems of quantum information theory. Discussing the links between entanglement and com- Negative information. The fact that sending qubits munication one has to mention about one more addi- means sending entanglement, can be used in even more tivity problem inspired by entanglement behavior. This complicated situation. Namely, suppose there is a noise- is the problem of additivity of quantum capacities Q less quantum channel from Alice to Bob, and Bob already touched already in (Bennett et al., 1996d). Due to non- has some partial information, i.e. now the source ρ is ini- additivity phenomenon called activation of bound en- tially distributed between Alice and Bob. The question tanglement (see Sec. XII.I) it has been even conjec- is: how many qubits Alice has to send to Bob, in or- tured (Horodecki et al., 1999a) that for some channels der he completes full information? First of all what does Q(Λ Λ ) > 0 even if both channels have vanishing ca- it mean to complete full information? This means that 1 ⊗ 2 pacities i.e. Q(Λ1) = Q(Λ2) = 0. Here again, as in the for all ensembles of ρ, after transmission, they should be 56 in Bob’s hands. Instead of checking all ensembles, one XV. QUANTIFYING ENTANGLEMENT can rephrase the above “completing information” in ele- gant way by means of entangled state: namely, we con- A. Distillable entanglement and entanglement cost sider purification ψABR of ρAB. The transmission of the quantum information lacking by Bob, amounts then just The initial idea to quantify entanglement was con- to merging part A of the state to Bob, without disturbing nected with its usefulness in terms of communication it. (Bennett et al., 1996c,d). As one knows via a two qubit 1 maximally entangled state ψ− = ( 01 10 ) one | i √2 | i − | i can teleport one qubit. Teleportation is a process that In (Horodecki et al., 2005h, 2006e) it was shown that involves the shared singlet, and local manipulations to- if classical communication is allowed between Alice and gether with communication of classical bits. Since qubits Bob then the amount of qubits needed to perform merg- cannot be transmitted by use of classical communication ing is equal to conditional entropy S(A B). This holds itself, it is clear that power of sending qubits should be even when S(A B) is negative, in which| case it means, attributed to entanglement. If a state is not maximally that not only Alice| and Bob do not need to send qubits, entangled, then it does not allow for faithful teleporta- but they get extra EPR pairs, which can be used for tion. However, in analogy to Shannon communication some future communication. What is interesting in the theory, it turns out that when having many copies in context of channels capacity, this result proves the Quan- such state, one can obtain asymptotically faithful tele- tum Shannon Theorem and hashing inequality. The new portation at some rate (see Sec. XII). To find how many proof is very simple (the protocol we have recalled in Sec. qubits per copy we can teleport it is enough to determine XII.F). how many e-bits we can obtain per copy, since every φ+ can then be used for teleportation. In this way we arrive| i at transition rates as described in Sec. XIII, and two Mother and father protocols. Several different phenom- basic measures of entanglement ED and EC . ena of theory of entanglement and quantum communi- Distillable entanglement. Alice and Bob start from n cation have been grasped by a common formalism dis- copies of state ρ, and apply an LOCC operation, that covered in (Devetak et al., 2004, 2005). There are two ends up with a state σn. We now require that for large n + mn main protocols called mother and father. In the first one, the final state approaches the desired state (Φ2 )⊗ . If the noisy resource is state, while in the second — chan- it is impossible, then ED = 0. Otherwise we say that the nel. From mother protocol one can obtain other proto- LOCC operations constitute a distillation protocol and mn P the rate of distillation is given by R = limn . The cols, such as dense coding via mixed states, or one-way P n distillation of mixed states (children). Similarly, father distillable entanglement is the supremum of such rates protocol generates protocols achieving quantum channel over all possible distillation protocols. It can be defined capacity as well as entangled assisted capacity of quan- concisely (cf. (Plenio and Virmani, 2006)) as follows tum channel. In both cases from children one can obtain n + ED(ρ) = sup r : lim inf Λ(ρ⊗ ) Φ2rn 1 =0 , parents, by using a new primitive discovered by Harrow n Λ k − k called cobit. n →∞ (148)o where 1 is the trace norm (see (Rains, 1998) for show- ing thatk·k other possible definitions are equivalent). 63 The discovery of state merging have given a new twist. Entanglement cost. It is a measure dual to ED, and In (Abeyesinghe et al., 2006) a different version of state it reports how many qubits we have to communicate in merging (called Fully Quantum Slepian Wolf protocol), order to create a state. This, again can be translated where only quantum communication and sharing EPR to e-bits, so that EC (ρ) is the number of e-bits one can pairsis allowed (and counted) was provided. It turns out obtain from ρ per input copy by LOCC operations. The that the minimal number of needed qubits is equal to definition is 1 I(A : R), and the number 1 I(A : B) of EPR pairsis 2 2 n + EC (ρ) = inf r : lim inf ρ⊗ Λ(Φ2rn ) 1 =0 . gained by Alice and Bob. The protocol is based on a n Λ →∞ k − k beautiful observation, that a random subsystem A′ of A n (149)o 1 of size less than 2 I(A : B) is always maximally mixed In (Hayden et al., 2001) it was shown, that EC is equal and product with reference R. This implies that after to regularized entanglement of formation EF — a proto- sending the second part of A to Bob, A′ is maximally en- type of entanglement cost, see Sec. XV.C.2. Thus, if EF tangled with Bob’s system. These ideas were further de- veloped in (Devetak and Yard, 2006), where operational meaning of conditional mutual information was discov- ered. It turned out that the above protocol is a powerful 63 In place of trace norm one can use fidelity, Uhlmann fi- primitive, allowing to unify many ideas and protocols. In delity F (ρ, σ) = (Trp√ρσ√ρ)2 thanks to inequality proven in 1 particular, this shows deep operational relations between (Fuchs and van de Graaf, 1997) 1 pF (ρ, σ) ρ σ 1 − ≤ 2 k − k ≤ entangled states and quantum channels. p1 F (ρ, σ). − 57 is additive, then the two quantities are equal. As we have called “separable operations” (Rains, 2000; Vedral et al., mentioned, for pure states ED = EC . 1997b) Distillable key. There is yet another distinguished op- erational measure of entanglement for bipartite states, Λ(ρ)= Ai Bi(ρ)Ai† Bi† (151) designed in similar spirit as ED and EC . It is distill- ⊗ ⊗ i able private key KD: maximum rate of bits of private X key that Alice and Bob can obtain by LOCC from state 64 ρAB where it is assumed that the rest E of the total pure with obvious generalization to multipartite setting . Ev- state ψABE is given to adversary Eve. The distillable key ery LOCC operation can be written in the above form, satisfies obviously ED KD: indeed, a possible protocol but not vice versa, as proved in (Bennett et al., 1999a). of distilling key is to≤ distill EPR states and then from (For more extensive treatment of various classes of oper- each pair obtain one bit of key by measuring in standard ations see Sec. XI.B.) basis. A crucial property of KD is that it is equal to the The known entanglement measures usually satisfy a rate of transition to a special class of states: so-called stronger condition, namely, they do not increase on av- private states, which are generalization of EPR states. erage We elaborate more on distillable key and the structure of private states in Sec. XIX. p E(σ ) E(ρ), (152) i i ≤ i X B. Entanglement measures — axiomatic approach where pi, σi is ensemble obtained from the state ρ by means{ of LOCC} operations. This condition was earlier The measures such as ED or EC are built to describe entanglement in terms of some tasks. Thus they arise considered as mandatory, see e.g. (Horodecki, 2001a; from optimization of some protocols performed on quan- Plenio, 2005), but there is now common agreement that tum states. However one can apply axiomatic point of the condition (150) should be considered as the only nec- 65 view, by allowing any function of state to be a mea- essary requirement . However, it is often easier to prove sure, provided it satisfies some postulates. Let us now the stronger condition. go through basic postulates. Interestingly, for bipartite measures monotonicity also implies that there is maximal entanglement in bipartite′ systems. More precisely, if we fix Hilbert space Cd Cd , 1. Monotonicity axiom then there exist states from which any other state can⊗ be created: these are states UA UB equivalent to singlet. The most important postulate for entanglement mea- Indeed, by teleportation, Alice⊗ and Bob can create from sures was proposed already in (Bennett et al., 1996d) still singlet any pure bipartite state. Namely, Alice prepares in the context of operationally defined measures. locally two systems in a joint state ψ, and one of systems teleports through singlet. In this way Alice and Bob Monotonicity under LOCC: Entanglement can- share the state ψ. Then they can also prepare any mixed • not increase under local operations and classical state. Thus E must take the greatest value to the state communication. φ+. Sometimes, one considers monotonicity under LOCC (Vedral et al., 1997b) introduced idea of axiomatic def- operations for which the output system has the same inition of entanglement measures and proposed that an local dimension as the input system. For example, for entanglement measure is any function that satisfies the n-qubit states, we are interested only in output n-qubit above condition plus some other postulates. Then (Vidal, states. Measures that satisfy such monotonicity can be 2000) proposed that monotonicity under LOCC should useful in many contexts, and sometimes it is easier to be the only postulate necessarily required from entan- prove such monotonicity (Verstraete et al., 2003), (see glement measures. Other postulates would then either section XV.H.1). follow from this basic axiom, or should be treated as op- tional (see (Popescu and Rohrlich, 1997) in this context). Namely, for any LOCC operation Λ we have 64 For more extensive treatment of various classes of operations, see E(Λ(ρ)) E(ρ). (150) ≤ Sec. XI.B. 65 Indeed, the condition (150) is more fundamental, as it tells about Note that the output state Λ(ρ) may include some regis- entanglement of state, while (152) says about average entangle- ment of an ensemble (family pi, ρi ), which is less operational ters with stored results of measurements of Alice and Bob { } (or more parties, in multipartite setting), performed in notion than notion of state. Indeed it is not quite clear what does it mean to “have an ensemble”. Ensemble can always be the course of the LOCC operation Λ. The mathematical treated as a state Pi pi i i ρi, where i are local orthogonal form of Λ is in general quite ugly (see e.g. (Donald et al., flags. However it is not| ih clear|⊗ at all why| i one should require a 2002)). A nicer mathematical expression is known as so priori that E(P pi i i ρi)= P piE(ρi). i | ih |⊗ i 58 2. Vanishing on separable states. negativity and all measures constructed by means of con- vex roof (see Sec. XV.C). It is open question whether If a function E satisfies the monotonicity axiom, it distillable entanglement is convex (Shor et al., 2001). In turns out that it is constant on separable states. It fol- multipartite setting it is known, that a version of distill- lows from the fact that every separable state can be con- able entanglement66 is not convex (Shor et al., 2003). verted to any other separable state by LOCC (Vidal, 2000). Even more, E must be minimal on separable states, because any separable state can be obtained by 4. Monotonicity for pure states. LOCC from any other state. It is reasonable to set this constant to zero. In this way we arrive at even more ba- For many applications, it is important to know whether sic axiom, which can be formulated already on qualitative a given function is a good measure of entanglement just level: on pure states, i.e. if the initial state is pure, and the final state (after applying LOCC operation) is pure. Entanglement vanishes on separable states • Since by Nielsen theorem (sec. XIII.A) for any pair of It is quite interesting, that the LOCC monotonicity states φ and ψ, one can transform φ into ψ iff φ is ma- axiom almost imposes the latter axiom. Note also that jorized by φ, one finds, that a function E is monotone on those two axioms impose E to be a nonnegative function. pure states if and only if it is Schur-concave as a function of spectrum of subsystem. I.e. whenever x majorizes y we have E(x) E(y). ≤ 3. Other possible postulates. Again, it is often convenient to consider a stronger monotonicity condition (152) involving transitions from The above two axioms are essentially the only ones that pure states to ensemble of pure states. Namely, it is should be necessarily required from entanglement mea- then enough to require, that any local measurement will sures. However there are other properties, that may be not increase entanglement on average (cf. (Linden et al., useful, and are natural in some context. Normalization. 1999a)). We obtain condition First of all we can require that entanglement measure behaves in an “information theoretic way” on maximally E(ψ) p E(ψ ), (155) ≥ i i entangled states, i.e. it counts e-bits: i X + n E((Φ2 )⊗ )= n. (153) where ψi are obtained from local operation + A slightly stronger condition would be E(Φd ) = log d. V ψ ψ = i . (156) For multipartite entanglement, there is no such a natural i V ψ condition, due to nonexistence of maximally entangled k i k state. Vi = Ai I (or Vi = I Bi) are Kraus operators Asymptotic continuity. Second, one can require some ⊗ ⊗ of local measurement (satisfying V †V = I) and p type of continuity. The asymptotic manipulations i i i i are probabilities of outcomes. The inequality should paradigm suggest continuity of the form (Donald et al., be satisfied for both Alice and BobP measurements. Re- 2002; Horodecki et al., 2000b; Vidal, 2000): call, that the notion of “measurement” includes also uni- E(ρn) E(σn) tary operations, which are measurements with single out- ρn σn 1 0 | − | 0, (154) comes. The condition obviously generalizes to multipar- k − k → ⇒ log d → n tite states. for states σ , ρ acting on Hilbert space of dimen- n n Hn sion dn. This is called asymptotic continuity. Measures which satisfy this postulate, are very useful in estimat- 5. Monotonicity for convex functions ing ED, and other transition rates, via inequality (210) (Sec. XV.E.2). The most prominent example of impor- For convex entanglement measures, the strong mono- tance of asymptotic continuity is that together with the tonicity under LOCC of Eq. (152) has been set in a above normalization and additivity is enough to obtain a simple form by Vidal (Vidal, 2000), so that the main unique measure of entanglement for pure states (see Sec. difficulty — lack of concise mathematical description of XV.E.1). LOCC operations — has been overcome. Namely, a con- Convexity. Finally, entanglement measures are often vex function f is LOCC monotone in the strong sense convex. Convexity used to be considered as a mandatory (152) if and only if it does not increase under ingredient of the mathematical formulation of monotonic- ity. At present we consider convexity as merely a conve- nient mathematical property. Most of known measures are convex, including relative entropy of entanglement, 66 E.g. maximal amount of EPR pairs between two chosen parties, entanglement of formation, robustness of entanglement, that can be distilled with help of all parties. 59 a) adding local ancilla C. Axiomatic measures — a survey f(ρ σ ) f(ρ ), X = A′,B′, (157) AB ⊗ X ≤ AB Here we will review bipartite entanglement measures b) local partial trace built on axiomatic basis. Some of them immediately gen- eralize to multipartite case. Multipartite entanglement f(ρAB) f(ρABX ), (158) measures we will present in Sec. XV.H ≤ c) local unitary transformations d) local von Neumann measurements (not necessarily 1. Entanglement measures based on distance complete), A class of entanglement measures (Vedral and Plenio, f(ρ ) p f(σi ), (159) 1998; Vedral et al., 1997b) are based on the natural in- AB ≥ i AB i tuition, that the closer the state is to the set of separable X i states, the less entangled it is. The measure is minimum where σAB is state after obtaining outcome i, and distance 67 between the given state and the states in : pi is probability of such outcome D S E , (̺) = inf (̺, σ). (163) Thus it is enough to check monotonicity under uni-local D S σ D operations (operations that are performed only on single ∈S The set is chosen to be closed under LOCC opera- site, as it was in the case of pure states). For a convex S function, all those conditions are equivalent to a single tions. Originally it was just the set of separable states one, so that we obtain a compact condition: S. It turns out that such function is monotonous un- For convex functions, monotonicity (152) is equivalent der LOCC, if distance measure is monotonous under all to the following condition operations. It is then possible to use known, but so far unrelated, results from literature on monotonicity under p E(σ ) E(ρ), (160) completely positive maps. Moreover, it proves that it i i ≤ i is not only a technical assumption to generate entangle- X 1 i i ment measures: monotonicity is a condition for a distance where the inequality holds for σi = W IBρW † IB, pi A A to be a measure of distinguishability of quantum states i i ⊗ i i ⊗ and p = Tr(W I ρW † I ), with W †W = I , i A ⊗ B A ⊗ B i A A A (Fuchs and van de Graaf, 1997; Vedral et al., 1997a). and the same for B (with obvious generalization to many We thus require that parties). P Convexity allows for yet another, very simple formula- (ρ, σ) (Λ(ρ), Λ(σ)) (164) tion of monotonicity. Namely, for convex functions strong D ≥ D monotonicity can be phrased in terms of two equalities and obviously (ρ, σ) = 0 for ρ = σ. This implies non- negativity of D(similarly as it was in the case of vanish- (Horodecki, 2005). Namely, a nonnegative convex func- D tion E is LOCC monotone (in the sense of inequality ing of entanglement on separable states). More impor- (152)), if and only if tantly, the above condition immediately implies mono- tonicity (150) of the measure E , . To obtain stronger [LUI] E is invariant under local unitary transforma- monotonicity, one requires pD S(̺ , σ ) (̺, σ). for • i iD i i ≤ D tions ensembles pi,̺i and qi, σi obtained from ρ and σ by { } { P} [FLAGS] for any chosen party X, E satisfies equality applying an operation. • Once good distance was chosen, one can consider dif- E p ρi i i = p E(ρ ), (161) ferent measures by changing the sets closed under LOCC i ⊗ | iX h | i i i i operations. In this way we obtain E ,PPT (Rains, 2000) X X D or E ,ND (the distance from nondistillable states). The where i are local orthogonal flags. D | iX measure involving set PPT is much easier to evaluate. The greater the set (see fig. 3), the smaller the measure 6. Invariance under local unitary transformations is, so that if we consider the set of separable states, those with positive partial transpose and the set of nondistill- Measure which satisfies monotonicity condition, is in- able states, we have variant under local unitary transformations E ,ND E ,PPT E ,S. (165) D ≤ D ≤ D E(ρ)= E(U ...U ρU † ...U † ). (162) 1 ⊗ N 1 ⊗ N In Ref. (Vedral and Plenio, 1998) two distances were Indeed these operations are particular cases of LOCC op- shown to satisfy (164) and convexity: square of Bu- erations, and are reversible. Thus monotonicity requires res metric B2 = 2 2 F (̺, σ) where F (̺, σ) = that E does not change under those operations. This − condition is usually first checked for candidate for en- p tanglement measures (especially if it is difficult to prove monotonicity condition). 67 We do not require the distance to be a metric. 60 [Tr(√̺σ√̺)1/2]2 is fidelity (Jozsa, 1994; Uhlmann, 1976) pure states. To see it, consider ̺ with optimal ensemble and relative entropy S(̺ σ) = Tr̺(log ̺ log σ). Origi- p , ψ . Consider local measurement with Kraus opera- | − { i i} nally, the set of separable states was used and the result- tors Vk. It transforms initial state ̺ as follows ing measure 1 ̺ qk, σk , qk = TrVk̺Vk†, σk = Vk̺Vk†. (168) ER = inf Tr̺(log ̺ log σ) (166) →{ } qk σ SEP − ∈ The members of the ensemble pi, ψi transform into en- is called relative entropy of entanglement. It is sembles of pure states (because{ operation} is pure) one of the fundamental entanglement measures, as the relative entropy is one of the most impor- i i i i 1 ψ q , ψ , q = Tr(V ψ ψ V †), ψ = V ψ . tant functions in quantum information theory (see i k k k k i i k k i k i →{ } | ih | qk (Schumacher and Westmoreland, 2000; Vedral, 2002)). (169) Its other versions - the relative entropy distance from 1 i i i p One finds that σk = piq ψ ψ . PPT states (Rains, 2000) and from nondistillable states qk i k| kih k| P P T ND Now we want to show that the initial entanglement (Vedral, 1998) - will be denoted as ER , and ER re- E(̺) is no less thanP the final average entanglement spectively. Relative entropy of entanglement (all its ver- E = q E(σ ), assuming, that for pure states E is sions) turned out to be powerful upper bound for entan- k k k monotonous under the operation. Since σk is a mixture glement of distillation (Rains, 2000). The distance based of ψi ’s,P then due to convexity of E we have on fidelity received interpretation in terms of Grover al- k gorithm (Shapira et al., 2005). 1 E(σ ) p qi E(ψi ). (170) k ≤ q i k k k i X 2. Convex roof measures Thus E p qi E(ψk). Due to monotonicity ≤ i i k k i on pure states qi E(ψi ) E(ψ ). Thus E Here we consider the following method of obtaining en- k k k i p E(ψ ).P However,P the ensemble≤ p , ψ was opti-≤ tanglement measures: one starts by imposing a measure i i i i i mal, so that the latterP term is equal simply{ to}E(̺). This E on pure states, and then extends it to mixed ones by P convex roof (Uhlmann, 1998) ends the proof. Thus, for any function (strongly) monotonous for pure states, its convex roof is monotonous for all states. As a E(̺) = inf p E(ψ ), p =1,p 0, (167) i i i i ≥ result, the problem of monotonicity of convex roof mea- i i X X sures reduces to pure states case. Let us emphasize that where the infimum is taken over all ensembles pi, ψi for the above proof bases solely on the convex-roof construc- { } tion, hence is by no means restricted to bipartite systems. which ̺ = i pi ψi ψi . The infimum is reached on a particular ensemble| ih (Uhlmann,| 1998). Such ensemble we In Sec. XV.D we will discuss the question of monotonic- call optimalP. Thus E is equal to average under optimal ity for pure states. In particular we will see that any ensemble. concave, expansible function of reduced density matrix The first entanglement measure built in this way of ψ satisfies (155). For example (Vidal, 2000) one can 1 α was entanglement of formation EF introduced in take Eα(ψ) given by Renyi entropy 1 α log2 Tr(̺ ) of − (Bennett et al., 1996d), where E(ψ) is von Neumann en- the reduction for 0 α . For α = 1 it gives EF , ≤ ≤ ∞ tropy of the reduced density matrix of ψ. It constituted while for α = 0 — the average logarithm of the number first upper bound for distillable entanglement. In Ref. of nonzero Schmidt coefficients. Finally, for α = we ∞ (Bennett et al., 1996d) monotonicity of EF was shown. obtain a measure related to the one introduced by Shi- In Ref. (Vidal, 2000) general proof for monotonicity of all mony (Shimony, 1995) when theory of entanglement did possible convex-roof measures was exhibited. We will re- not exist: call here the latter proof, in the form (Horodecki, 2001a). One easily checks that E is convex. Actually, con- E(ψ)=1 sup ψ ψprod , (171) − ψprod |h | i| vex roof measures are the largest functions that are (i) convex (ii) compatible with given values for pure states. where the supremum is taken over all product pure states Now, for convex functions, there is the very simple con- ψprod (see Sec. XV.H.1 for multipartite generalizations dition of Eq. (160) equivalent to strong monotonicity. of this measure). Namely, it is enough to check, whether the measure does We will consider measures for pure bipartite states not increase on average under local measurement (with- in more detail in Sec. XV.D, and multipartite in Sec. out coarse graining i.e. where outcomes are given by XV.H.1. Kraus operators). Using this condition, we will show, that if a measure is monotone on pure states (according to Eq. (155)), then a. Schmidt number The Schmidt rank can be extended its convex roof extension is monotonous on mixed states. to mixed states by means of convex roof. A differ- Thus the condition (152) is reduced to monotonicity for ent extension was considered in (Sanpera et al., 2001; 61 Terhal and Horodecki, 2000) (called Schmidt number) as it is very simple for pure states. Namely C2 is a polyno- follows mial function of coefficients of a state written in standard basis rS (̺) = min(max[rS (ψi)]), (172) i C(ψ)=2 a00a11 a01a10 , (178) where minimum is taken over all decompositions ̺ = | − | pi ψi ψi and rS(ψi) are the Schmidt ranks of the for ψ = a 00 + a 01 + a 10 + a 11 . i | ih | 00 01 10 11 corresponding pure states. Thus instead of average One can extend| thisi quantity| i to higher| i dimensions| i P Schmidt rank, supremum is taken. An interesting fea- (Audenaert et al., 2001c; Rungta et al., 2001) (see also ture of this measure is that its logarithm is strongly (Badzi¸ag et al., 2002)) by building concurrence vector. nonadditive. Namely there exists a state ρ such that The elements Cα of the vector are all minors of rank rS (ρ)= rS(ρ ρ). two: ⊗ ψ ψ ψ ψ Cα(ψ)=2(Aij Ai′j′ Aij′ Ai′j ) (179) b. Concurrence For two qubits the measure called − concurrence was introduced for pure states in ψ where ψ = ij Aij ij , α = (ii′, jj′) with i Let λ1,...,λ4 be singular valuesp of ω in decreasing order. One finds that Then we have C(ρ) = inf (U UBU U ) (182) ⊗ † ⊗ † mm,mm C(ρ) = max 0, λ1 λ2 λ3 λ4 (175) m { − − − } X q Interestingly Uhlmann has shown (Uhlmann, 2000) that where B is biconcurrence matrix 75. This matrix can for any conjugation Θ, i.e. antiunitary operator sat- m µ ( ) ′ ′ − ′ 1 be written as follows Bmµ,nν = ψAB ψA B PAA isfying Θ = Θ− , the convex roof of the function Θ- ( ) n ν h |h | ⊗ P − ′ ψ ψ ′ ′ , from which one can show that it is concurrence CΘ(ψ)= ψ Θ ψ is given by generalization BB | ABi| A B i of Wootters’ formula: h | | i positive, hence it can be written in eigendecomposition as B = α χα χα , with unnormalized, orthogonal χα. d It was shown| thatih | C (ρ) = max 0, λ λ , (176) P Θ { 1 − i} i=2 C(ρ) max λ λ , 0 (183) X ≥ { 1 − i } λi>1 where λi are eigenvalues of operator √ρ√ΘρΘ in decreas- X ing order. where λ are singular values of operator z Aχα put in The importance of the measure stems from the fact i α α decreasing order, with zα being arbitrary chosen complex that it allows to compute entanglement of formation for numbers satisfying z 2 and χ =P Aχα ij . A two qubits according to formula (Wootters, 1998) α α α ij ij very good computable bound| | is obtained already| ifi the P P 1+ 1 C2(ρ) linear combination α zαχα is reduced to a chosen single EF (ρ)= H( − ), (177) χα. In the case of two qubits B is of rank one, hence we p 2 obtain the WoottersP result. where H is binary entropy H(x) = x log x (1 There are other interesting measures introduced in x) log(1 x). Another advantage of concurrence− − is that− (Fan et al., 2003; Sino l¸ecka et al., 2002) and developed − 62 in (Gour, 2005), which are built by means of polynomi- the feature of nonincreasing on average under local mea- als of the Schmidt coefficients λ’s: surements, so that it is LOCC monotone. If we start with mutual information, we obtain measure interpolat- d ing between entanglement of formation and squashed en- τ1 = λi =1, tanglement introduced in (Tucci, 2002) and later inde- i=1 X pendently in (Nagel and Raggio, 2005). Its monotonicity d under LOCC was proved later (Horodecki et al., 2006b). τ2 = λiλj , Let us mention, that if initial function f is asymptoti- i>j X cally continuous, then both usual convex roof, as well as d mixed convex roof are asymptotically continuous too (it τ3 = λiλj λk, was shown in (Synak-Radtke and Horodecki, 2006), gen- i>j>kX eralizing the result of Nielsen (Nielsen, 2000) for EF ). . . (184) 4. Other entanglement measures The above measures τp are well defined if the dimension a. Maximal teleportation fidelity/maximal singlet fraction. of the Hilbert space d is no smaller than the degree p. For a state ρ one can consider fidelity of teleportation For convenience, one can also set τp = 0 for p < d, so of a qudit averaged uniformly over inputs and maxi- that each of the above measures is well defined for all mized over trace preserving LOCC operations. Denote pure states. The functions are generalizations of concur- it by fmax. It is related to maximal fidelity Fmax with rence and can be thought as higher level concurrences. In d d maximally entangled state Φ+ = 1 ii opti- ⊗ | i i √d | i particular τ2 is square of concurrence. The measures are mized over trace preserving LOCC (Rains, 1999) as fol- P Schur-concave (i.e. they preserve majorization order), so lows (Horodecki et al., 1999b) that by Nielsen theorem (see sec. XIII.A) they satisfy monotonicity. dFmax +1 Due to simplicity of concurrence measure, an interest- fmax = . (186) d +1 ing quantitative connection has been found with comple- mentarity between visibility and which-path information Both quantities are by construction LOCC monotones. in interference experiments (Jakob and Bergou, 2003). They are not equal to zero on separable states, but are Interesting generalizations of concurrence were found in constant on them: Fmax(ρsep)=1/d. For two qubits, the multi-partite case (see Sec. XV.H) protocol to obtain Fmax is of the following simple form (Verstraete and Verschelde, 2003): Alice applies some fil- ter Ai I (see Sec. XI), and tells Bob whether she suc- 3. Mixed convex roof measures ceeded⊗ or not. If not, then they remove the state and 1 produce some separable state which has overlap d with One can consider some variation of convex roof method singlet state. by allowing decompositions of a state into arbitrary Another measure was constructed in (Brandao, 2005) states rather than just pure ones (Horodecki et al., by means of activation concept (Horodecki et al., 1999a). 2006b). This allows to produce entanglement measures It is connected with the maximal fidelity with Ψ+, ob- from other functions that are not entanglement measures tainable by means of local filtering, denoted here by themselves. Namely, consider function f which does not (p) Fmax (i.e. probabilistic maximal singlet fraction, see sec. increase on average under local measurement (recall, that XII.I). Namely consider a state σ having some value we mean here generalized measurement, so that it in- (p) Fmax(σ). With help of some other state ρ one can ob- cludes unitary transformations). I.e. f satisfies the con- (p) (p) dition (160), so that if only it were convex, it would be tain better Fmax, i.e. Fmax(ρ σ) may be larger than (p) ⊗ monotone. However, what if the function is not convex? Fmax(σ). What is rather nontrivial, this may happen (p) Example of such quantity is quantum mutual informa- even if ρ is bound entangled, i.e. for which Fmax is the tion. It is obviously not a monotone, because e.g. it same as for separable states. The activation power of a takes different values on separable states. given state can be quantified as follows From such function we could obtain entanglement mea- sure by taking usual convex roof. However, we can also (p) (p) (d) Fmax(ρ σ) Fmax(σ) take mixed convex roof defined by E (ρ) = sup ⊗(p) − . (187) σ Fmax(σ) E(ρ) = inf p f(ρ ), (185) i i Since Masanes (Masanes, 2005a) showed that any entan- i X gled state can activate some other state, i.e. increase (p) (d) where infimum is taken over all decompositions ρ = its Fmax, the quantity E is nonzero for all entangled i piρi. The new function is already convex, and keeps states, including all bound entangled states. P 63 b. Robustness measures Robustness of entanglement was infW :TrW =1(TrρW ), Rg = infW I( TrρW ), Ebsa = − − ≤ − introduced in (Vidal and Tarrach, 1999). For a state ρ infW I(TrρW ). − ≥ consider separable state σsep. Then R(ρ σsep) is defined Interestingly, the singlet fraction maximized over PPT as minimal t such that the state | operations (Rains, 2000) can be represented as: 1 ppt 1 (ρ + tσsep) (188) Fmax = inf TrW ρ, (194) 1+ t d − W is separable. Now robustness of entanglement is defined 1 where the infimum is taken over the set W : (1 d) I { − ≤ as W 1 I, 0 W Γ 2 1 I . ≤ d ≤ ≤ d } R(ρ) = inf R(ρ σsep). (189) σsep | e. Negativity. A simple computable measure was intro- It is related to the quantity P (ρ) given by minimal p such duced in (Zyczkowski˙ et al., 1998) and then shown in that the state (Vidal and Werner, 2002) to be LOCC monotone. It is negativity (1 p)ρ + pσ (190) − sep = λ (195) N is separable. We have P = R/(1 + R). Though P is λ<0 more intuitive, it turns out that R has better mathemat- X ical properties, being e.g. convex. R satisfies monotonic- where λ are eigenvalues of ρΓ (where Γ is partial trans- ity (152). In (Harrow and Nielsen, 2003; Steiner, 2003) pose). A version of the measure called logarithmic nega- generalized robustness Rg was considered, where the in- tivity given by fimum is taken over all states rather than just separable Γ ones. Interestingly it was shown that for pure states it E (ρ) = log ρ 1 (196) N k k does not make difference. Rg is monotone too. Brandao (Brandao, 2005) showed that the generalized robustness is upper bound for distillable entanglement (Vidal and Werner, 2002). It can be also written Rg has operational interpretation: it is just equal to the 2 (ρ)+1 (d) measure E quantifying the activation power. More- as E (ρ) = log N 2 . The measure E (ρ) is easily seenN to satisfy monotonicity (150), becauseN (ρ) does over (Brandao, 2005) Rg gives rise to the following upper satisfy it, and logarithm is monotonic function.N However bound for ED: logarithm it is not convex, and as such might be expected ED(ρ) log2(1 + Rg(ρ)). (191) not to satisfy the stronger monotonicity condition (152). ≤ However it was recently shown that it does satisfy it (Plenio, 2005). The measure is moreover additive. For Γ Γ c. Best separable approximation measure was introduced states with positive ρ E has operational interpreta- | | N in (Karnas and Lewenstein, 2000) by use of best separa- tion — it is equal to exact entanglement cost of creating ble approximation idea (Lewenstein and Sanpera, 1998). state by PPT operations from singlets (Audenaert et al., The latter is defined as follows: one decomposes state ρ 2003). as a mixture It turns out that negativity, robustness, BSA can be also obtained from one scheme originating form base ρ = (1 p)δρ + pσsep, (192) norm (Plenio and Virmani, 2006; Vidal and Werner, − 2002) where σsep is separable state, and δρ is arbitrary state. Denote by p∗ the maximal possible p. Now, it turns out that Ebsa(ρ)=1 p∗ is an entanglement measure, i.e. f. Greatest cross norm. In Ref. (Rudolph, 2001) a mea- it is LOCC monotone− and vanishes on separable states. sure based on the so called greatest cross-norm was pro- For all pure entangled states the measure is equal to 1. posed. One decomposes ρ into sum of product operators ρ = A B . (197) i ⊗ i i d. Witnessed entanglement In (Brandao, 2005; X Brandao and Vianna, 2006) entanglement measures Then the measure is given by are constructed using entanglement witnesses as follows: E(ρ) = sup A B , (198) E = inf TrρW, (193) k ik1 ·k ik1 − W i X where infimum is taken over some set of entanglement where supremum is taken over all decompositions (197). witnesses. It turns out that many measures can be recast It is not known if the function satisfies monotonicity (in in this form. For example random robustness is equal to (Rudolph, 2001) monotonicity under local operations and 64 convexity was shown). However it was shown (Rudolph, and EC . Even though it was computed for just two fam- 2005) that if in infimum one restricts to Hermitian oper- ilies of states, a clever guess for ρABE can give very good ators, then it is equal to 2R + 1, where R is robustness estimates for ED in some cases. of entanglement. It is an open question if the optimization in the defini- tion of Esq can be restricted to the subsystem E being a classical register (Tucci, 2002). If it is so, then we could g. Rains bound Rains (Rains, 2000) has combined two express Esq as follows: different concepts (relative entropy of entanglement and ? i negativity) to give the following quantity Esq = inf piIM (ρAB ), (203) i Γ ER+ = inf S(̺ σ)+ σ 1 , (199) X N σ | k k where the infimum is taken over all decompositions i where the infimum is taken under the set of all states. i piρAB = ρAB. (This measure was called c-squashed entanglement, because it can be recast as a version of It is the best known upper bound on distillable entan- P glement. However it turns out to be an entanglement Esq with E being a classical register). Esq would be then measure itself. One can show that the measure satisfies nothing else but a mixed convex roof measure based on monotonicity (152). To see this, consider optimal σ (i.e. mutual information mentioned above, which is an entan- reaching infimum), the existence of which is assured by glement measure itself. continuity of norm and lower semi-continuity of relative entropy ((Ohya and Petz, 1993)). For such a σ by mono- tonicity of relative entropy under any quantum operation i. Conditioning entanglement It is worth to mention that and monotonicity of E under LOCC operation we have there is a method to obtain a new entanglement measure for any LOCC map Λ N from a given one as follows (Yang et al., 2007): Γ Γ CE(ρAB) = inf (E(ρAA′BB′ ) E(ρA′B′ )), (204) ′ ′ S(̺ σ)+ σ 1 S(Λ(̺) Λ(σ)) + Λ(σ) 1. (200) ρAA BB − | k k ≥ | k k Hence the infimum can not be increased under LOCC. where infimum is taken over all states ρAA′BB′ such that TrA′B′ ρAA′BB′ = ρA′B′ . Provided the initial measure was convex and satisfied strong monotonicity, the same h. Squashed entanglement. Squashed entanglement was holds for the new measure. Moreover it is automatically introduced by (Tucci, 2002) and then independently by superadditive, and is a lower bound for regularization Christandl and Winter (Christandl and Winter, 2003), E∞ of the original measure. who showed that it is monotone, and proved its additiv- ity. It is therefore the first additive measure with good asymptotic properties. In the latter paper, definition of D. All measures for pure bipartite states squashed entanglement Esq has been inspired by rela- tions between cryptography and entanglement. Namely, In Ref. (Vidal, 2000) it was shown that measures for pure states satisfying strong monotonicity (152) are in Esq was designed on the basis of a quantity called intrin- sic information (Gisin and Wolf, 2000; Maurer and Wolf, one-to-one correspondence to functions f of density ma- 1997; Renner and Wolf, 2003), which was monotonic un- trices satisfying der local operations and public communication. The (i) f is symmetric, expansible function of eigenvalues squashed entanglement is given by of ̺ 1 (ii) f is concave function of ̺ Esq (ρAB) = inf I(A : B E) (201) ρABE 2 | (by expansibility we mean f(x1,...,xk, 0) = where I(A : B E) = S + S S S and infi- f(x ,...,x )). In this way all possible entanglement | AE BE − E − ABE 1 k mum is taken over all density matrices ρABE satisfying measures for pure states were characterized. TrEρABE = ρAB. The measure is additive on tensor More precisely, let Ep, defined for pure states, satisfy product and superadditive in general, i.e. Ep(ψ)= f(̺A), where ̺A is reduction of ψ, and f satis- fies (i) and (ii). Then there exists an entanglement mea- Esq(ρAB ρA′B′ )= Esq(ρAB)+ Esq(ρA′B′ ); sure E satisfying LOCC monotonicity coinciding with ⊗ Esq (ρAA′BB′ ) Esq(ρAB)+ Esq(ρA′B′ ). (202) Ep on pure states (E is convex-roof extension of Ep). ≥ Also conversely, if we have arbitrary measure E satisfy- It is not known whether it vanishes if and only if ing (152), then E(ψ) = f(̺A) for some f satisfying (i) the state is separable. (It would be true, if infi- and (ii). mum could be turned into minimum, this is however We will recall the proof of the direct part. Namely, unknown). The measure is asymptotically continuous we will show that convex-roof extension E of Ep satisfies (Alicki and Fannes, 2004), and therefore lies between ED (152). As mentioned earlier, it suffices to show it for pure 65 states. Consider then any operation on, say, Alice side If we fix an entangled state φ0, then p(ψ φ0) is (for Bob’s one, the proof is the same) which produces itself a measure of entanglement, as a function of→ψ. The ensemble p , ψ out of state ψ. We want to show that relation (206) gives in particular { i i} the final average entanglement E = piE(ψi) does not i E (ψ) exceed the initial entanglement E(ψ). In other words, we p(ψ φ) k (207) (i) P (i) need to show p f(̺ ) f(̺ ), where ̺ are reduc- → ≤ Ek(φ) i i A ≤ A A tions of ψi on Alice’s side. We note that due to Schmidt P for all k. It turns out that these are the only constraints decomposition of ψ, reductions ̺A and ̺B have the same for transition probabilities. Namely (Vidal, 1999) The non-zero eigenvalues. Thus, f(̺A) = f(̺B), due to (i). optimal probability of transition from state ψ to φ is given (i) (i) Similarly f(̺A )= f(̺B ). Thus it remains to show that by (i) (i) i pif(̺B ) f(̺B). However ̺B = i pi̺B (which is ≤ Ek(ψ) algebraic fact, but can be understood as no-superluminal- p(ψ φ) = min . (208) Psignaling condition — no action on AliceP side can influ- → k Ek(φ) ence statistics on Bob’s side, provided no message was This returns, in particular, Nielsen’s result (Nielsen, transmitted from Alice to Bob). Thus our question re- 1999). Namely, p = 1, if for all k, E (ψ) E (φ), which (i) (i) k ≥ k duces to the inequality pif(̺ ) f( pi̺ ). This is precisely the majorization condition (120). i B ≤ i B is however true, due to concavity of f. Thus the considered set of abstractly defined measures As we have mentioned,P examples of entanglementP mea- determines possible transformations between states, as sures for pure states are quantum Renyi entropies of sub- well as optimal probabilities of such transformations. We system for 0 α 1. Interestingly, the Renyi entropies will further see generalization of such result to asymptotic for α> 1 are≤ not≤ concave, but are Schur concave. Thus regime, where nonexact transitions are investigated. they are satisfy monotonicity (150) for pure states, but do not satisfy the strong one (152). It is not known whether the convex roof construction will work, there- E. Entanglement measures and transition between states fore it is open question how to extend such measure to — asymptotic case mixed states. Historically first measure was the von Neumann en- In asymptotic regime, where we tolerate small inac- tropy of subsystem (i.e. α = 1) which has operational curacies, which disappear in the limit of large num- interpretation — it is equal to distillable entanglement ber of copies, the landscape of entanglement looks more and entanglement cost. It is the unique measure for pure “smooth”. Out of many measures for pure states only states, if we require some additional postulates, especially one becomes relevant: entropy of entanglement, i.e. any asymptotic continuity (see Sec. XV.E). measure significant for this regime reduces to entropy of entanglement for pure states. 68 Moreover, only the measures with some properties, such as asymptotic con- tinuity, can be related to operational quantities such as 1. Entanglement measures and transition between states — ED, or more generally, to asymptotic transitions rates. exact case We refer to such measures as good asymptotic measures. Another family of entanglement measures is the follow- ing. Consider squares of Schmidt coefficients of a pure 1. ED and EC as extremal measures. Unique measure for pure state λk in decreasing order so that λ1 λ2 ...λd, bipartite states. d ≥ ≥ i=1 λi = 1. Then the sum of the d k smallest λ’s − If measures satisfy some properties, it turns out that P d their regularizations are bounded by ED from one side Ek(ψ)= λi (205) and by EC from the other side. By regularization of 1 n i=k any function f we mean f (ρ) = lim f(ρ ) if such X ∞ n n ⊗ a function exists. It turns out that if a function E is is an entanglement monotone (Vidal, 1999). Thus for monotonic under LOCC, asymptotically continuous and + a state with n nonzero Schmidt coefficients we obtain satisfies E(ψd ) = log d, then we have n 1 nontrivial entanglement measures. It turns out − that these measures constitute in a sense a complete set ED E∞ EC . (209) ≤ ≤ of entanglement measures for bipartite pure states. Vidal proved the following inequality relating proba- bility p(ψ φ) of obtaining state φ from ψ by LOCC 68 with entanglement→ measures: One can consider half-asymptotic regime, where one takes limit of many copies, but does not allow for inaccuracies. Then other measures can still be of use, such as logarithmic negativity which E(ψ) p(ψ φ) . (206) is related to PPT-cost of entanglement (Audenaert et al., 2003) → ≤ E(φ) in such regime. 66 In particular this implies that for pure states, there is This result is an asymptotic counterpart of Vidal’s rela- unique entanglement measure in the sense that regular- tion between optimal probability of success and entan- ization of any possible entanglement measure is equal to glement measures (206). Noting that, in particular, we entropy of subsystem69. An exemplary measure that fits have this scheme is relative entropy of entanglement (related + + 1 either to the set of separable states or to PPT states). R(ρ ψ )= ED(ρ), R(ψ ρ)= , (211) → 2 2 → E (ρ) Thus whenever we have reversibility, then the transition C rate is determined by relative entropy of entanglement. one easily arrives at extreme measures theorem. For Some versions of the theorem are useful to find upper example to obtain bounds for transition rates between bounds for ED - one of central quantities in entangle- maximally correlated we can choose measures ER and ment theory. We have for example that any function E EF , as they satisfy the conditions and are additive for satisfying the following conditions: those states. For more sophisticated transitions see n (Horodecki et al., 2003b). One notes that R gives rise 1. E is weakly subadditive, i.e. E(ρ⊗ ) nE(ρ), ≤ to plethora entanglement measures, since the following d functions 2. For isotropic states E(ρF ) 1 for F 1, d , log d → → → ∞ 1 ED(ρ)= R(ρ σ), EC (ρ)= , (212) σ → σ R(σ ρ) 3. E is monotonic under LOCC (i.e. it satisfies Eq. → (150)), where σ is an entangled state, are nonincreasing under LOCC. (Thanks to the fact that every entangled state is an upper bound for distillable entanglement. This the- have nonzero E the above measures are well defined.) orem covers all known bounds for E . C D Upper bound on distillable key. This paradigm allows There are not many measures which fit the asymptotic to provide upper bound K (see Sec. XIX.B.5), because regime. apart from operational measures such as E , D C distillable key is also some rate of transition under LOCC E and K only entanglement of formation, relative en- D D operations. Namely, any entanglement measure which tropy of entanglement (together with its PPT version) is asymptotically continuous, and for so-called private and squashed entanglement belong here. For review of state γ satisfies E(γ ) log d, then it is upper bound properties of those measures see (Christandl, 2006). d d ≥ for KD (Christandl, 2006). Roughly speaking it follows from the fact, that distillable key can be expressed as rate of transition 2. Transition rates KD(ρ) = sup R(ρ γd)log d, (213) One can consider transitions between any two states γd → (Bennett et al., 1996d) by means of LOCC: R(ρ σ) → where supremum is taken over all p-dits γd (see Sec. defined analogously to ED, but with σ in place of max- XIX.B.2). imally entangled state. Thus we consider n copies of ρ out and want to obtain a state σn that will converge m copies of σ for large n. R(ρ σ) is then defined as F. Evaluating measures maximum asymptotic rate m/n→that can be achieved by LOCC operations. One then can generalize the theorem It is usually not easy to evaluate measures. The only about extreme measures as follows: measure that is easily computable for any state is E (logarithmic negativity). Entanglement of formationN is E (ρ) R(ρ σ) ∞ (210) efficiently computable for two-qubits (Wootters, 1998). → ≤ E∞(σ) Other measures are usually computable for states with high symmetries, such as Werner states, isotropic state, for any E satisfying: or the family of “iso-Werner” states, see (Bennett et al., 1. E is nonincreasing under LOCC, 1996d; Rains, 1999, 2001; Terhal and Vollbrecht, 2000; Vollbrecht and Werner, 2001). 2. regularizations exist for states ρ and σ and An analytical lower bounds for concurrence for all E∞(σ) > 0, states were provided in (Mintert et al., 2004) (see also (Audenaert et al., 2001c)). The bound constitutes also a 3. E is asymptotically continuous (see Sec. XV.B.3). new criterion of separability. A way to bound a convex roof measure, is to provide a computable convex func- tion, that is greater than or equal to the measure on the pure states. For example we have 69 Uniqueness of entanglement measure for pure states was put for- 2 ward in (Popescu and Rohrlich, 1997). The postulates that lead to uniqueness were further worked out in (Donald et al., 2002; ( ψ ψ )Γ = R( ψ ψ ) = √p (214) k | ih | k1 k | ih | k1 i Horodecki et al., 2000b; Vidal, 2000). i ! X 67 where pi are squares of Schmidt coefficients of ψ, and R For higher dimension powerful tools for evaluating ED is realignment map VI.B.8. Comparing this with concur- were provided in (Rains, 2000). One knows several upper rence one gets a bound obtained in (Chen et al., 2005a) bounds for ED (Bennett et al., 1996d; Horodecki et al., 2000a; Rains, 1999, 2000; Vedral and Plenio, 1998; Vidal and Werner, 2002). The best known bound is 2 Γ C(ρ) (max( ρ 1, R(ρ) 1) 1). (215) ER+ provided in (Rains, 2000). For Werner states it is ≥ sm(m 1) k k k k − N − equal to regularization of ER, (this is true for more gen- S As far as entanglement of formation is concerned, in eral class of symmetric states (Audenaert et al., 2001b)). (Terhal and Vollbrecht, 2000) a method was introduced Squashed entanglement has been evaluated for so for which it is enough to optimize over some restricted set called flower state (purification of which is given by rather than the set of pure states. This was further suc- (272)) and its generalizations in (Christandl and Winter, cessfully developed in (Chen et al., 2005b; Datta et al., 2005). 2006a; Fei and Li-Jost, 2006) where lower bounds for EF Several entanglement measures have been evaluated were obtained based on known separability criteria such for some multipartite pure and mixed graphs states in as PPT, realignment or the recent Breuer’s map (see (Markham et al., 2007). They have used results con- Secs. VI.B.8 and VI.B.6). cerning distinguishing states via LOCC (276), see sec. In (Vollbrecht and Werner, 2001) a surprising result XVIII.B. was obtained, concerning possible additivity of ER. They have shown that ER is nonadditive for Werner asymmet- ric states, and moreover for large d, ER of two copies is almost the same as for one copy. Thus, the relative entropy of entanglement can be strongly nonadditive. 1. Hashing inequality Therefore regularization of ER is not equal to ER. In Any entanglement measure E that is good in asymp- (Audenaert et al., 2001a) for the first time ER∞ was com- puted for some states. Namely for Werner states we have: totic regime satisfy inequality E S(̺B) S(̺AB) coh ≥ − ≡ IA B. For EF the inequality follows from concavity of 1 H(p) 1 0. It seems that we can have ED = EC only for states of the form pi ψi ψi AB i i A′B′ (217) | ih | ⊗ | ih | 2. Evaluating EC vs additivity problem i X where i ′ ′ are product states distinguishable by Alice It is long open question of whether in general E is | iA B F and Bob (Horodecki et al., 1998b) (or some generaliza- additive. Since ED is lower bound for EC , it follows that tions in similar spirit, that Alice and Bob can distinguish for distillable states EC is nonzero. Though for bound states, which satisfy ED = EC trivially). entangled states (for which ED = 0) this does not give Apart from the above trivial case of locally orthogo- any hint. In (Vidal and Cirac, 2001) the first example of nal mixtures the value of the measure ED is known only bound entangled state with EC > 0 was found. Later for maximally correlated states a ii jj for which (Vidal et al., 2002) examples of states with additive E ij ij | ih | F ED = SA SAB. It is upper bound, since it is equal to were found. − P ER (Rains, 1999). That it can be achieved follows from Example. We will now show a simple proof (see general result of (Devetak and Winter, 2005) stating that (Horodecki et al., 2005d)), that for any state of the form ED SA SAB. Example is mixture of two maximally entangled≥ − two qubit states where we have ρ = a ii jj (219) AB ij | ih | ED(̺)=1 S(̺). (218) ij − X 68 EF is additive. Such states are called maximally corre- of ρAA′:BB′ into pure states (which gives mixed convex 70 lated . Namely, one considers purification ψABC which roof on the right hand side — as a result, EF remains can be taken as ii ψ˜ where ψ˜ ψ˜ = a . E , but C changes into G). i | iAB| iiC h i| j i ij F HV The state of subsystems B and C is of the form ρBC = Now we apply this last inequality to n copies of the P ˜ i pi i i ψi ψi , where ψi are normalized vectors ψi same state ρ and obtain | ih | ⊗ |2 ih | and pi = aii . Then using the fact that EF can be also | | (n 1) (n 1) Pdefined as infimum of average entanglement between Al- EF (ρ⊗ − ρ) EF (ρ⊗ − )+ G(ρ). (225) ⊗ ≥ ice and Bob over all measurements performed by Charlie on system C one finds that Iterating this equation we obtain n E = S I ( p , ψ ) (220) EF (ρ⊗ ) EF (ρ) + (n 1)G(ρ). (226) F B − acc { i i} ≥ − 71 where Iacc is so called accessible information . Now, Dividing both sides by n we obtain EC G(ρ). It re- following Wootters (see (DiVincenzo et al., 2002)), one mains to prove that G is nonzero for any entangled≥ state. can easily show by use of chain rule that Iacc is additive. This follows from standard arguments of Caratheodory’s Since SB is additive too, we obtain that EF is additive. type. More generally we have (Koashi and Winter, 2004) E (A : B)= S(B) C (C B) (221) F − HV i G. Entanglement imposes different orderings where CHV is a measure of classical correlations (Henderson and Vedral, 2001). It is defined as follows: One can ask, whether different entanglement measures impose the same ordering in set of states. The question i was first posed in (Virmani and Plenio, 2000). Namely, CHV (A B) = sup S(ρB ) piS(ρB ) (222) i Ai − suppose that E(ρ) E(σ). Is it also the case that { } i ≥ X E′(ρ) E′(σ)? That it is not the case, we can see just ≥ where supremum is taken over all measurements Ai on pure states. There exist incomparable states, i.e. such {i } on system A, and pi is probability of outcome i, ρB is states ψ, φ, that neither ψ φ nor φ ψ is possible by the state of system B, given the outcome i occurred. In LOCC. Since LOCC transitions→ is governed→ by entangle- (Devetak and Winter, 2004a) it was shown that for sepa- ment measures (see Sec.XV.D) we see that there are two rable states CHV is additive. This is related to the result measures which give opposite ordering on those states. by Shor (Shor, 2002b) on additivity of classical capacity In asymptotic regime there is unique measure for for entanglement breaking channels (actually the original pure states. However, again it is easy to see results on additivity of EF were based on Shor result). (Virmani and Plenio, 2000), that a unique order would Another result on EC was provided in (Yang et al., imply ED = EC for all states, while we know that it is 2005a). Namely, EC is lower bounded by a function G not the case. which is (mixed) convex roof of the function CHV . Such On can interpret this lack of single ordering as follows: a function is nonzero if and only if a state is entangled. there are many different types of entanglement, and in It is worth to present a proof, as it is quite simple, and one state we have more entanglement of one type, while again uses duality between EF and CHV . in other state, there is more entanglement of some other We first consider a pure state ψAA′BB′ . For this state type (See (Miranowicz, 2004b; Verstraete et al., 2004b) we have in this context). E (ψ ′ ′ )= S(AA′)= E (ρ ′ )+ C (B′ AA′) F AA :BB F AA :B HV i ≥ E (ρ )+ C (B′ A′) (223)H. Multipartite entanglement measures ≥ F A:B HV i where ρXXX denote suitable partial traces. Many of the axiomatic measures, are immediately ex- Now consider mixed state ρAA′:BB′ . We obtain tended to multipartite case. For example relative en- tropy of entanglement is generalized, by taking a suit- E (ρ ′ ′ ) E (ρ )+ G(B′ A′), (224) F AA :BB ≥ F A:B i able set in place of bipartite separable states. One can take the set of fully separable states (then the measure where G is mixed convex roof of CHV . This follows from applying left hand side to optimal decomposition will not distinguish between “truly multipartite” entan- glement and several instances of bipartite entanglement + + 72 such as φAB φCD) . To analyze truly multipartite entanglement,⊗ one has to consider as in (Vedral et al., 70 Additivity was shown in (Horodecki et al., 2003b) along the lines of Ref. (Vidal et al., 2002). 71 It is defined as follows: Iacc( pi, ψi ) = sup Aj I(i : j) where { } { } 72 supremum is taken over all measurements, and I(i : j) is mutual Some inequalities between so chosen version of ER and bipartite information between symbols i and measurement outcomes j. entanglement were provided in (Plenio and Vedral, 2000) 69 1997b) the set of all states containing no more than k- The first measure that is neither easy combination of particle entanglement (see Sec. VII). Similarly one can bipartite measures, nor an obvious generalization of such proceed with robustness of entanglement. It is not easy a measure is 3-tangle (or residual tangle) introduced in however to compute such measures even for pure states (Coffman et al., 2000). It is defined as follows (see e.g. (Plenio and Vedral, 2000)). Moreover, for mul- τ(A : B : C)= τ(A : BC) τ(AB) τ(AC), (227) tipartite states, much more parameters to describe en- − − tanglement is needed, therefore many new entanglement where 2-tangles on the right hand side are squares of measures have been designed, especially for pure states. concurrence (175). 3-tangle is permutationally invariant, Then they can be extended to all states by convex roof even though the definition does not suggest it. It may (which is however also hard to compute). be though zero for pure states that are 3-entangled (i.e. Before we present several such measures, let us make that are not product with respect to any cut). Exam- a digression on behavior of k-party entanglement (see ple is so called W state. Tangle vanishes on any states Sec. VII) under tensor product. Consider two states, that are separable under any cut, and is nonzero for ex- φ+ 0 and φ+ 0 . Both are 2 entangled: they AB C AC B ample on GHZ state.74 There are attempts to define a contain| i | noi three-partite| i | entanglement.i But if Alice, Bob good generalization of tangle for multiqubit systems by and Charlie have both states, they can create e.g. GHZ means of hyper-determinant (Miyake, 2003) (see below). state. It thus follows, that the ”k-partyness” of entangle- In (Lohmayer et al., 2006) a convex roof of 3-tangle was ment is not closed under tensor product. One of implica- computed for mixture of GHZ state and W state orthog- tions is that distillable-GHZ entanglement is extremely onal to it. superadditive: it jumps from zero to 1 upon tensoring Shortly after introducing tangle, a concept of an- the above states. other measure for tripartite states was introduced in the The above discussion concerns scenario where we ten- context of asymptotic rate of transitions (Linden et al., sor systems such a way that the number of parties does 1999a): not increase: i.e. ψ ψ ′ ′ ′ = ψ ′ ′ . This ABC ⊗ A B B AA :BB :CC way of interpreting tensoring is characteristic for informa- E(ψ)= ER(ρAB )+ S(ρC) (228) tion theoretic applications, being analogous to many uses of channel. It seems that for analysis of physical many- where ρAB,ρC are reductions of ψABC . The measure body systems, it is more natural to interpret tensoring allowed to detect truly tripartite entanglement in GHZ as adding new parties: ψABC ψDEF = ψABCDEF . In state in asymptotic regime (see sec. XIII.B.3). It is easy such case, “k-partyness” of entanglement⊗ is preserved un- to see that it is monotonic under Alice and Bob actions. der tensoring. Namely, the term ER(ρAB) represents entanglement of There is even more subtle notion of ”genuinely mul- ρAB (hence does not increase under Alice and Bob ac- tipartite entanglement” for pure states, which is defined tions) and the term S(ρC ) represents entanglement of the recursively as follows. An entangled bipartite pure states total state under the cut AB : C (hence does not increase has always genuine bipartite entanglement. A m-partite under action of any party). However the first term can pure state has genuine m-partite entanglement if its (m- be usually increased by Charlie. Still the whole measure 1)-particle reduced density matrices can be written as is monotone, because this increase is always accompanied mixtures of pure states which do not have genuinely (m- by a larger decrease of the term S(ρC). Thus to increase 1)-partite entanglement. For three qubits it is 3-tangle, entanglement between A and B, one has to use up entan- which is nonzero iff the state is genuinely 3-partite en- glement between AB : C. Equivalently ER(ρA:B) goes tangled. It is then in GHZ class. 73 down under mixing less than the entropy goes up. Any measure having this feature can be put to the above for- mula, to create a new entanglement measure. However, 1. Multipartite entanglement measures for pure states it is known that entanglement of formation will not work here, because it is lockable, i.e. it can decrease arbitrar- There are measures that are simple functions of sums ily under loss of one bit, i.e. increase entropy by 1 (see of bipartite entanglement measures. Example is “global section XVIII.A). entanglement” of (Meyer and Wallach, 2001) which is One of the first measures designed specifically for mul- sum of concurrences between single qubit versus all other tipartite states was Schmidt measure (Eisert and Briegel, qubits. Their monotonicity under LOCC is simply inher- 2001). This is minimum of log r where r is number ited from bipartite measures. of terms in an expansion of the state in product basis. For GHZ this measure is 1, because there are just two 73 Let us emphasize, that there is a different notion of genuine multiparty entanglement in asymptotic domain. There a pure 74 In turns out, that if tangles in (227) are replaced by squares of state contains genuinely triparty entanglement if it cannot be negativities the obtained quantity (after symmetrizing over sys- reversibly transformed into EPR pairs (see sec. XIII.B.3). It is tems permutation) gives also rise to an entanglement monotone then not known, whether W state is of this sort or not. (Ou and Fan, 2007). 70 terms: 000 and 111 . One can show, that for W state on homogeneous functions described above. Explicit for- it is impossible| i to| writei it by means of less than three mula for hyperdeterminant for four qubits can be found terms (otherwise it would either belong to GHZ class, or in (Levay, 2006). to EPR class). The measure is zero if and only if the Geometric measure. A family of measures have been state is fully product. Therefore, it cannot distinguish defined by Barnum and Linden (Barnum and Linden, truly multipartite entanglement from bipartite entangle- 2001). In particular so called geometric measure is de- ment. However it may be useful in many contexts, see fined as eg. (Mora and Briegel, 2005). E (ψ)=1 Λk(ψ), (230) An interesting general class of multipartite entan- g − glement measures was obtained in the context of where Λk(ψ) = sup ψ φ 2, with S being set of classification of states via so called normal forms φ Sk k k-separable states. This∈ |h is| generalizationi| of Shimony (Verstraete et al., 2003). Namely, consider any homoge- measure (Shimony, 1995), which for bipartite states was neous function of the state. Then if it is invariant under related to Renyi entropy with α = . For relations determinant one SLOCC i.e. it satisfies with robustness of entanglement see (Cavalcanti,∞ 2006). The measure was also investigated in (Wei et al., 2004; f(A1 ...Anψ)= f(ψ) (229) ⊗ Wei and Goldbart, 2003) where it was in particular com- for Ai being square matrices satisfying det Ai = 1, puted for Smolin four qubit bound entangled states (85). then it is entanglement monotone in strong sense (152), Concurrence-like measures. There were other at- but under the restriction that the LOCC operation tempts to generalize concurrence. Christensen and Wong produces output states on the Hilbert space of the same (Wong and Christensen, 2001) obtained a measure for dimension. The 3-tangle is example of such a measure. even number of qubits by exploiting conjugation that Many measures designed for pure multipartite states like appeared in original definition of concurrence for two those obtained in (Akhtarshenas, 2005; Miyake, 2003; qubits. Their concurrence works for even number of n Osterloh and Siewert, 2005; Wong and Christensen, qubits and is given by ψ∗ σy ψ . The measure is nonzero 2001) are originally defined only for a fixed dimension for a four partite statesh two| | pairsi of EPR states. This hence it is simply not possible to check the standard approach was generalized in (Osterloh and Siewert, 2005, monotonicity (150). However concurrence though 2006) who analysed systematically possible quantities initially defined for qubits, can be written in terms of built out of antilinear operations, also of higher order linear entropy of subsystems, being thus well defined in ψ than concurrence. For example, they obtained the for all systems. Therefore there is a hope, that one following nice representation for 3-tangle: can arrive at definition independent of dimension for µ τ = ψ σ σ σ ψ∗ ψ σ σ σ ψ∗ (231) other measures. Then to obtain full monotonicity, one h | µ ⊗ y ⊗ y| ih | ⊗ y ⊗ y| i will need in addition to prove, that the measure does where µ = 0, 1, 2, 3 and the contraction is described not change, if the state is embedded into larger Hilbert by tensor gµ,ν = diag[ 1, 1, 0, 1]. They have also de- spaces of subsystems (equivalently, that the measure signed measures that distinguish− between three different does not change under adding local ancilla). However, SLOCC classes of states (see sec. XIII.A.2): for four-qubit concurrence of (Wong and Christensen, 4 2001) ψ∗ σy ψ its natural generalization was shown to be noth monotonous| | i (Demkowicz-Dobrza´nski et al., 2006) 1 Φ1 = ( 0000 + 1111 ) (see below). Of course, even the functions that are only | i √2 | i | i monotonous for fixed dimension are useful quantities in 1 many contexts. Φ2 = (√2 1111 + 1000 + 0100 + 0010 + 0001 ) | i √6 | i | i | i | i | i Measures based on hyperdeterminant. Miyake noticed 1 that measures of entanglement such as concurrence and Φ3 = ( 1111 + 1100 + 0010 + 0001 ). (232) tangle are special cases of hyperdeterminant (Miyake, | i 2 | i | i | i | i 2003). Consider for example qubits. For two qubits An interesting proposal is due to Akhtarshenas concurrence is simply modulus of determinant, which is (Akhtarshenas, 2005), which however is not proved to hyperdeterminant of first order. Tangle is hyperdetermi- be a monotone. In (Demkowicz-Dobrza´nski et al., 2006; nant of second order — a function of tensor with three Mintert et al., 2005b) a family of functions of the form indices. Though computing hyperdeterminants of higher order than tangle is rather complex, basing just on prop- C ( ψ )=2 ψ ψ ψ ψ (233) erties of hyperdeterminant Miyake proved, that hyperde- A | i h | ⊗ h |A| i ⊗ | i p was introduced, where = ps1...s Ps1 ...Ps , terminants of higher degree are also entanglement mono- A s1...sn n ⊗ n tones (Miyake, 2004). They describe truly multipartite with s = 1, P ( 1) is projector onto symmetric (an- i ± P entanglement, (in a sense, that states such as product of tisymmetric)± subspace (see Sec. VI.B.3), and the coeffi- EPR’s have zero entanglement). The proof of monotonic- cients ps1...sn are nonnegative. They have given sufficient ity bases on geometric-arithmetic mean, and is closely re- conditions that must be satisfied by the coefficients to en- lated to the construction of entanglement measures based sure monotonicity of C (now without the restriction of 71 fixed dimension). On the other hand they have shown by cooperation between Alice, Bob and Charlie ( ) ( ) ( ) ( ) that if = P − P − P − P − then the function (Gour and Spekkens, 2005). It is also nonadditive A ⊗ ⊗4 ⊗ returns concurrence ψ∗ σy ψ , and it is not monotonic. (DiVincenzo et al., 1998a). However in the limit of many The main tool was theh following| | i condition for monotonic- copies, it turns out that it becomes a monotone, namely ity derived on basis of conditions LUI and FLAGS see sec. it reduces to minimum of entropies of Alice’s and Bob’s XV.B.5). Namely a function C that subsystem (Smolin et al., 2005), and for larger number of parties to minimum entropy over all cuts that divide is real, nonnegative and invariant under local uni- Alice from Bob. (This was generalized to N parties in • taries (Horodecki et al., 2005h, 2006e).) satisfies C(a ψ )= a 2C( ψ ) • | i | | | i is defined for mixed states as a convex roof I. Entanglement parameters • is an entanglement monotone if and only if C(a ψ Some quantities even though are not monotonic un- η + b φ η ) a 2C(a ψ η )+ bC( φ | ηi ⊗) 1 1 1 1 der LOCC, seem still useful in quantitative descrip- with| i equality| i ⊗ | fori a≤= | | 0 or |b =i ⊗ 0, | wherei ψ |andi ⊗φ | arei tion of entanglement. For example the parameter arbitrary multipartite pure states, and η , η are local 1 2 M(ρ) (Horodecki et al., 1995) reporting maximal viola- orthogonal flags. tion of Bell-CHSH inequality for two-qubit states, or Multipartite version of squashed entanglement. We can N(ρ) reporting maximal fidelity of teleportation for a obtain multipartite versions of both squashed and c- class of protocols (Horodecki et al., 1996b) (see Sec. squashed entanglement, by putting in place of mutual in- IV). One of the most important quantities in quan- formation of bipartite system, its generalization for mul- tum communication theory, coherent information, in- tipartite systems (Horodecki et al., 2006b) troduced in (Schumacher and Nielsen, 1996) (see also I(A1 : . . . : AN )= S(A1)+ . . . + S(AN ) S(A1 ...AN ), (Lloyd, 1997)), can be positive only if the state is en- − (234) tangled (Horodecki and Horodecki, 1994). One feels that 1 the greater are such quantities, the more entangled is choosing normalization factor N , to get 1 on GHZ state, and applying the conditioning rule S(X Y )= S(XY ) the state. However those quantities can increase under S(Y ). So constructed multipartite entanglement| mea-− LOCC. For example coherent information can increase sure is closely related to multipartite secrecy monotones even under local partial trace. introduced in (Cerf et al., 2002b). Thus they cannot describe entanglement directly, as In the context of spin chains, another measure of it would imply, that entanglement can be increased by entanglement was designed: localisable entanglement means of LOCC. However, it is plausible that the above (Verstraete et al., 2004a). Namely, one chooses two parameters simply underestimate some measures. Let us spins, and performs LOCC operations aiming at ob- consider coherent information in more detail. One can taining the largest bipartite entanglement between them consider maximum value of coherent information attain- (measured according to a chosen entanglement measure able by LOCC. This is already an entanglement measure. for two bipartite states). Localisable entanglement is One can show, that this value does not exceed log d, i.e. a generalization of entanglement of assistance, initially the value on singlet state. Thus coherent information can defined for tripartite pure states in (DiVincenzo et al., only underestimate the value of entanglement measure. 1998a) as maximal entropy of entanglement that can be Let us note that it is important to know the maximal created between Alice and Bob, if Charlie helps them by value of the entanglement measure induced by the given measuring his system and telling the outcomes. For tri- parameter, so that we have a reference point. partite pure states entanglement of assistance is simply a function of bipartite state ρAB of Alice and Bob (how- ever it reflects entanglement properties of joint state, not J. How much can entanglement increase under communication of one qubit? ρAB). Its formula is dual to entanglement of formation: instead of infimum, there is supremum In (Lo and Popescu, 1999) it was postulated that un- i Eass(ρ) = sup piS(ρA), (235) der sending n qubits entanglement shouldn’t increase pi,ψi { } i more than by n. In (Chen and Yang, 2000) it was shown X for entanglement of formation. Due to teleportation, i where ρA is reduced density matrix of ψi, and supremum sending qubits is equivalent to bringing in a singlet. The is taken over all decompositions of ρ. For example con- question can then be recast as follows: which entangle- sider GHZ state. Alice and Bob themselves share clas- ment measures satisfy sically correlated states. However, Charlie can measure in basis + , , and if he tells them result, they obtain E(ρ φ+ φ+ ) E(ρ)+ n. (236) | i |−i ⊗ | ih | ≤ EPR pair. So Eass = 1 in this case. Entanglement of assistance is not a monotone (Of course it is meaningful to ask such questions only for tripartite states, because it could be increased for those entanglement measures that exhibit a sort of 72 2 2 2 extensive behavior.) If a measure is subadditive, i.e. One finds that CA:B = CAC = 1, while CA:BC =3/4. E(ρ σ) E(ρ)+ E(σ) then the condition is satisfied. Interestingly the monogamy for concurrence implies ⊗ ≤ This is the case for such measures as ER, EF , EC , EN , monogamy of negativity (see Sec.XV.B.4) (Ou and Fan, Esq. More problematic are ED and KD. As far as ED is 2007) concerned, it is easy to see that (236) is satisfied for distil- lable states. Simply, if by adding singlet, we can increase 2 + 2 2 . (240) NA:B NA:C ≤ NA:BC ED, then we could design protocol that would produce more singlets than ED, by using singlets obtained from It is not known if this holds in higher dimension. distilling a first bunch of copies to distillation of the next More generally in terms of entanglement measures bunch75. A more rigorous argument includes continuity monogamy takes the following form: of ED on isotropic states (singlets with admixture of ran- For any tripartite state of systems A1, A2, B dom noise). It is not hard to see, that for PPT states the condition also holds (it is actually enough to check it on E(A : B)+ E(A : C) E(A : BC). (241) ≤ two qubit singlet): Note that if the above inequality holds in general (i.e. E (ρ φ+ φ+ ) E (ρ φ+ φ+ ) not only for qubits), then it already itself implies (by D ⊗ | ih | ≤ N ⊗ | ih | = E (ρ)+ E ( φ+ φ+ ). (237) induction) the inequality N N | ih | This was later shown 76 for all states, by exploiting results E(A : B1)+ E(A : B2)+ . . . + E(A : BN ) of (DiVincenzo et al., 2003b). The question is still open E(A : B1 ...BN ). (242) ≤ for KD. In (Koashi and Winter, 2004) it was shown that squashed entanglement satisfies this general monogamy XVI. MONOGAMY OF ENTANGLEMENT E (A : B)+ E (A : C) E (A : BC). (243) sq sq ≤ sq One of the most fundamental properties of entangle- ment is monogamy (Coffman et al., 2000; Terhal, 2000b). This is the only known entanglement measure hav- In its extremal form it can be expressed as follows: If ing this property. EF and EC are not monogamous two qubits A and B are maximally quantumly correlated (Coffman et al., 2000; Koashi and Winter, 2004) e.g. on they cannot be correlated at all with third qubit C. In the above Aharonov state. This somehow support a view, general, there is trade-off between the amount of entan- that EC does not say about entanglement contents of glement between qubits A and B and the same qubit A the state, but rather describes entanglement that must and qubit C. This property is purely quantum: in clas- be “dissipated” while building the state by LOCC out of sical world if A and B bits are perfectly correlated, then pure entanglement (Horodecki et al., 2002; Synak et al., there is no constraints on correlations between bits A and 2005). (It is also supported by the locking effect, see Sec. C. For three qubits the trade-off is described by Coffman- XVIII.A). In the context of monogamy, one can consider Kundu-Wootters monogamy inequality other tradeoffs similar to (241). For example we have (Koashi and Winter, 2004) C2 + C2 C2 , (238) A:B A:C ≤ A:BC E (C : A)+ C (B A)= S , (244) F HV i A where CA:B is the concurrence between A and B, CA:C where CHV is a measure of classical — between A and C, while CA:BC — between system A and BC. There was a conjecture that the above in- (Henderson and Vedral, 2001) given by 222. equality can be extended to n-qubits. The conjecture has This says that if with one system we share too much en- been proved true only recently (Osborne and Verstraete, tanglement, this must suppress classical correlations with 2006). The monogamy is also satisfied for Gaussian the other system. One might think, that since classical states (Adesso and Illuminati, 2006; Hiroshima et al., correlations are suppressed, then also quantum should, 2007) see sec. XVII.E. However, it does not hold any- as existence of quantum correlations imply existence of more in higher dimension (Ou, 2006). Namely consider classical ones. Qualitatively it is indeed the case: no clas- Aharonov state of three qutrits sical correlations means that a state is product, hence cannot have entanglement too. Quantitatively the issue 1 is a bit more complicated, as EF can be greater than ψABC = ( 012 + 120 + 210 021 210 102 ). √6 | i | i | i−| i−| i−| i CHV . However, if other prominent measures of entangle- (239) ment would be smaller than CHV , we could treat it as an oddity of EC as mentioned above. To our knowledge it is not known, whether ER, ED, KD are smaller than CHV . Let us note that this latter relation would imply 75 D. Gottesman, private communication monogamy for those measures (since they are all upper 76 P. Shor, A. Harrow and D. Leung, private communication bounded by EF ). 73 A beautiful monogamy was found for Bell inequal- An example of entangled state from such space is two- ities. Namely, basing on the earlier results concern- mode squeezed state: which has its l2 l2-like representa- ing a link between the security of quantum commu- tion (in so called Fock basis considered⊗ to be a standard nication protocols and violation of Bell’s inequalities one): (Scarani and Gisin, 2001a) and theory of nonlocal games (Cleve et al., 2004), Toner proved the CHSH inequal- ∞ Ψ = 1 λ2 λn n ,n (247) ity is monogamous (Toner, 2006). This means, that if | λi − | i| i n=0 three parties A,B and C share quantum state ρ and each p X chooses to measure one of two observables then trade-off where index goes from zero for physical reasons (n rep- between AB’s and AC’s violation of the CHSH inequality resents the photon number). Here coefficients an := is given by (1 λ2)λn are just Schmidt coefficients. − 2 2 Tr( AB ρ) + Tr( AC ρ) 4. (245) Alternatively the state has its L L representation: | BCHSH | | BCHSH |≤ ⊗ It shows clearly that CHSH correlations are monogamous 2 2r 2 2r 2 i.e. if AB violate the CHSH inequality, then AC cannot. Ψλ(q1, q2)= exp[ e− (q1+q2) /2 e (q1 q2) /2], rπ − − − This is compatible with the present ideas of drawing cryp- (248) tographic key from nonlocality, where monogamy of non- related to previous representation by local correlations is the main feature allowing to bound the knowledge of eavesdropper. λ = tanh(r). (249) There is a qualitative aspect of monogamy, recog- nized quite early (Doherty et al., 2005; Werner, 1989b). In the case of infinite squeezing r the Ψ(q1, q2) be- comes more and more similar to the→ delta ∞ function δ(q Namely, a state ρAB is separable if and only if for any 1 − N there exists its N + 1 partite symmetric extension, i.e. q2) while its Fourier transform representation (chang- ing “positions” qi into “momenta” pi) becomes almost state ρAB1...BN , such that ρABi = ρAB. D. Yang in a re- cent paper (Yang, 2006) has provided an elegant proof of δ(p1 + p2). This limiting case was originally discussed in this result and has given an explicit bound on the number the famous EPR paper (Einstein et al., 1935), and per- of shared in terms of quantity G (see Sec. XV.F.2) which fect correlations in positions as well as momenta resemble is a mixed convex roof from Henderson-Vedral measure us perfect correlations of local measurements σx and σz on sides of the two-qubit state Ψ = 1 ( 0 0 + 1 1 ). of classical correlations, and itself is indicator of entan- + √2 | i| i | i| i glement, in a sense that it is zero if and only if a state is The separability property are in case of bipartite CV entangled. Namely we have pure states easy — as in discrete case we have for bipar- tite pure states equivalence S N A . (246) ≤ G(A B) separability PPT criterion i ⇔ reduced state pure For pure states SA = G(A B) which shows that no sym- ⇔ metric extension is possible,i if only state is entangled, i.e. Schmidt rank one (250) G = 0. ⇔ 6 The entropy of entanglement of pure states remains a good measure of entanglement, exhibiting however some XVII. ENTANGLEMENT IN CONTINUOUS VARIABLES oddities. In the case of the state (247) it is given by SYSTEMS (Giedke et al., 2003a; Wolf et al., 2004) 2 2 2 2 A. Pure states EF (Ψλ) = cosh (r)log2(cosh (r)) sinh (r)log2(sinh (r)). − (251) Many properties of entanglement (separability) change However in bipartite case with both subsystems of CV when passing to continuous variables since the infinite type typically the entropy of entanglement is infinite. dimensional Hilbert space is not compact. This is a consequence of the fact that generically quan- The term continuous variables comes from the fact that tum states on CV spaces have the entropy infinite (Wehrl, any infinite dimensional Hilbert space is isomorphic to 1978) (or alternatively the set of density matrices with any of the two spaces: finite entropy is nowhere dense i.e. contains no ball). 2 As an example of such state take ΨAB = (i) l ( ) which is space of sequences Ψ = ci with C 2 { } n √pn n A n B, with pn proportional (up to normal- i∞=1 ci < and scalar product Ψ Φ = | i | i | | ∞ h | i ization factor) to 1/(n + 2)log(n + 2)4. Then entropy of i∞=1 ai∗bi) and P P entanglement is infinite, since the series n pn log pn is (ii)P space L2(R) of all functions Ψ : R with not convergent. → C Ψ(x) 2(x)dx < and scalar product defined As one can expect the very important factP connected to R | | ∞ as Ψ(x) Φ(x)dx The variable x is just a contin- it is that there is no maximally entangled state in such R R ∗ uous variable (CV) here. spaces. Simply the state with all Schmidt coefficients R 74 equal does not mathematically exists (it would have in- There exist nontrivial PPT states finite norm). (Horodecki and Lewenstein, 2000) that cannot be A natural question here arises: what about usefulness constructed as a naive, direct sum of finite dimensional of infinite entanglement which is so common phenomenon structures. It seems that such states are generic, though in CV? For example is it possible to distill infinite amount the definition of generic CV state in case of mixed state of two-qubit entanglement from single copy of bipartite is not so natural as in case of pure state where infinite CV quantum states with infinite entanglement? The Schmidt rank (or rank of reduced density matrix) is a answer to this is negative and can be proven formally natural signature defining CV property (for discussion (Keyl et al., 2002). of the property of being generic see (Horodecki et al., Also, one can easily see that even for pure states EF 2003e)). is not continuous (Eisert et al., 2002b). Consider the se- The first important observation concerning CV sepa- quence of states rability is (Clifton and Halvorson, 2000) that in bipar- tite case the set of separable states is nowhere dense n 1 1 or — equivalently — any state on this space is a limit Ψn = 1 e f + e f | ABi − n log n| 0i| 0i n log n | ii| ii (in trace norm) of sequence of entangled state. Thus r r i=1 X (252) set of separable states contains no ball of finite radius and in that sense is “of zero volume” unlike it was in it is elementary to see that the sequence converges ˙ to a state with zero entanglement, Ψ Ψ finite dimensions (Zyczkowski et al., 1998). This result k| ABih AB| − can be extended (Horodecki et al., 2003e) to the set of e0 e0 f0 f0 1 0, while its entanglement diverges (|Eih(Ψ|n ⊗) | ih ).k → all nondistillable states (in a sense of definition inherited F AB → ∞ from discrete variables i.e. equivalent to impossibility of producing two-qubit singlets) is also nowhere dense in B. Mixed states set of all states. Thus CV bound entanglement like CV separability is a rare phenomenon. The definition of mixed separable states has to be Let us pass to quantitative issues involving entangle- changed slightly if compared to discrete variables: the ment measures. If one tries to extend definition of en- state is separable if it is a limit (in trace norm) of finite tanglement of formation to mixed states (Eisert et al., convex combination of, in general mixed and not pure, 2002b) then again set of states with finite EF has the product states77: same property as the set of separable states — it is again nowhere dense.78 Also, it is not continuous (as we have ̺sep p(n)̺(n),i ̺(n),i 0. (253) already seen in the case of pure states). k AB − i A ⊗ B k1 → i The question was how to avoid, at least partially, the X above problems with entanglement that occur when both The characterization of entanglement in terms of posi- dimensions are infinite? The authors of (Eisert et al., tive maps and entanglement witnesses is again true since 2002b) propose then to consider subset (H) SM ⊂ S the corresponding proofs are valid for general Banach ( set of all bipartite states defined as M (H) := ̺ : spaces (see (Horodecki et al., 1996a)). Tr(S ̺H) < M for some fixed constant MS and Hamilto-{ There is a small difference here: since there is no max- nian H (some} chosen Hermitian operator with spectrum imally entangled state one has to use the original version bounded from below). The set is nowhere dense but it of Jamio lkowski isomorphism between positive maps and is defined by natural physical requirement of bounded entanglement witnesses: mean energy in a physical system. Remarkably for fixed M and all states from M , the entanglement of forma- Λ ′ S WAB = (I Λ)(VAA ), (254) tion E and the relative entropy of entanglement E are ⊗ F R continuous in trace norm on pure states. Moreover those with d d (remember that it may happen that only A ≥ B measures are asymptotically continuous on pure states of one of subspaces is infinite) where V ′ is a swap operator n AA the form σ⊗ with finite-dimensional support of σ. on ′ = ′ , ( ′ being a copy of ) and the HAA HA ⊗HA HA HA map acting from system A′ to B. This is because there is no maximally entangled state in infinite dimensional C. Gaussian entanglement space. The PPT criterion is well defined and serves as a There is a class of CV states that are very well char- separability criterion as it was in finite dimension. It acterized with respect to separability. This is the class has a very nice representation in terms of moments (Shchukin and Vogel, 2005a). 78 This problem does not occur when one of the local Hilbert spaces A, B is finite i.e. min[dA, dB ] < then entanglement of 77 This is actually the original definition of separable states formationH H is well defined and restricted∞ by logarithm of the di- (Werner, 1989a). mension of the finite space (Majewski, 2002). 75 of Gaussian states. Formally a Gaussian state of m criterion (see (Werner and Wolf, 2001a)): modes (oscillators) is a mixed state on Hilbert space 2 m ̺ (Gaussian separable (260) (R)⊗ (of functions of ξ = [q1,...,qm] position vari- ⇔ Lables) which is completely characterized by the vector γ (̺) γ γ (261) AB ≥ A ⊕ B of its first moments di = Tr(̺Ri) (called displacement vector) and second moments covariance matrix γij = for some variance matrices γA,γB (which, as further was Tr(̺ Ri diI), (Rj dj I +), where we use anticommu- shown (Simon, 2003), can be chosen to be pure, i.e. of tator{ , − and the− cannonball} observables are position the form (259)). Quite remarkably the above criterion { }+ Qk = R2k and momentum Pk = R2k+1 operators of can be generalized to an arbitrary number of parties (see k-th oscillator which satisfy the usual Heisenberg com- (Eisert and Gross, 2005)). In general — if the state is not mutation relations [R , R ′ ] = iJ ′ where J = n J , Gaussian the criterion becomes only a necessary condi- k k kk ⊕i=1 i 0 1 tion of separability. The criterion is rather hard to use with one mode symplectic matrices J = . i 1− 0 (see however (Werner and Wolf, 2001a) and the discus- Any general matrix S is called symplectic iff it satisfies sion below). SJST = J. They represent all canonical transformations The PPT separability criterion takes a very easy form T for Gaussians. On the level of canonical variables par- S : ξ ξ′ = Sξ where ξ = [q1,p1; q2,p2; . . . ; qmpm] is a vector→ of canonical variables. The corresponding action tial transpose with respect to the given subsystem (say on the Hilbert space is unitary. There is also a broader B) corresponds to reversal of its conjugate variables ξ = [ξ , ξ ] ξ˜ = Λ where Λ = diag[I , σz ]. If we set of unitary operations called quasifree or linear Bogol- A B → A B ubov transformations ξ Sξ + d where S is symplectic. introduce transformation The canonical operators→ Q , P are real and Hermitian i i X˜ =ΛXΛ, (262) part of the creation ak† and annihilation ak operators that 2 m provide a natural link to (l )⊗ representation since they then we have PPT criterion for Gaussian states: bipartite distinguish a special l2 Fock basis n of each mode {| i} Gaussian state is PPT iffγ ˜ is physical i.e. satisfies either (a† = ∞ √n +1 n + 1 n and a = (a† )†) via of the conditions (256)-(258). Note that the first one k n=0 | ikkh | k k number operator N = a† a = ∞ n n n which is satisfied byγ ˜ can be written as: P k k n=0 diagonal in that basis. | ih | P γ + iJ˜ 0. (263) Since displacement d can be easily removed by ≥ quasifree local (i.e. on each mode separately) unitary There is a very important separability characteriza- operations (Duan et al., 2000)), only the properties of tion: PPT criterion has been shown to be both neces- variance matrix are relevant for entanglement tests. Be- sary and sufficient for 1 1 (Duan et al., 2000; Simon, fore recalling them we shall provide conditions for γ to 2000) and subsequently generalized× to 1 n Gaussians be physical. Let us recall that via Williamson theo- (Werner and Wolf, 2001a). Further the same× result has rem γ can be diagonalised with some symplectic matrix T been proven for m n “bisymmetric” (Serafini, 2006; γdiag = SV γSV = diag[κ1,κ1; . . . ; κm,κm], with κi real. Serafini et al., 2005)× (i.e. symmetric under permutations The physical character of the variance matrix γ is guar- of Alice and Bob modes respectively) Gaussian states. anteed by the condition: The equivalence of PPT condition to separability is not γ + iJ 0 (255) true, in general, if both Alice and Bob have more than T≥1 ⇔ one mode. In particular an example of 2 2 Gaus- γ J γ J (256) × ≥ ⇔ sian bound entanglement has been shown by proving (via γ ST S (257) ≥ ⇔ technique similar to that from (Lewenstein and Sanpera, κ 1,i =1,...,m. (258) 1998)) that the following covariance matrix i ≥ There is a remarkable fact(Simon, 2000): Gaussian state 20 0 010 0 0 is pure if and only if its variance matrix is of the form 01 0 000 0 1 − T 00 2 000 1 0 γ = S S, (259) − 00 0 10 10 0 γ = − (264) for some symplectic matrix S. Generally, given m modes 10 0 020 0 0 can be divided into k groups containing m1,...,m2,mk 00 0 104 0 0 − modes (m = mi), belonging to different local ob- 0 0 1000 2 0 i − servers A1,...,Ak. We say that the state is a k-partite 0 10 000 0 4 P − Gaussian state of m1 m2 mk type. For instance × ×···× the bipartite state is of m1 m2 type iff the first m1 does not satisfy the condition (261). modes are on Alice side, and× the rest m on Bob one. The PPT test for 1 1 Gaussian states can be 2 × All reduced states of the systems Ai are Gaussians. With written elementary if we represent the variance as each site we associate the symplectic matrix J as be- A C Ak γ = . Then the two elementary condi- fore. There is general necessary and sufficient separabil- CT B ity condition that can resemble to some extent the range tions (Simon, 2000) det(A)det(B) +(1/8 det(C))2 ± − 76 Tr(AJCJBJCT J)/8 + (det(A) + det(B))/8 0 repre- variance and becomes then necessary and sufficient sep- sent the physical and PPT character of γ respectively.≥ arability criterion (and hence equivalent to PPT) in this These conditions can be further simplified if the vari- two mode case (Duan et al., 2000). Further, it is interest- ance is driven by local linear unitaries (corresponding to ing to note that the PPT separability criterion implies a symplectic operations) to canonical form of type I, where series of uncertainty principle like relations. For any n n × matrices A and B are proportional to identities and C is mode state the physical condition γAB + iJ 0 and ob- diagonal (Simon, 2000). An interesting and more fruitful servables equivalent to the statement that≥ observables ~~ ~ ~ for general (nongaussian) case approach is offered by un- X(d) = dξ, X(d′) = d′ξ obey the uncertainty relation certainty relations approach where the so called type II is (see (Simon, 2003)) achieved by local quasi free unitary operations and PPT 2 2 ~ ~ ~ ~ condition is represented by some uncertainty relation (see (∆X(d)) ̺ + (∆X(d′)) ̺ dAJAdA′ + dBJBdB′ , h i h i ≥ | | (Duan et al., 2000) and subsequent section). (266) ~ ~ ~ ~ ~ ~ In (Giedke et al., 2001c) the separability problem for for any real vectors d = (dA, dB), d′ = (dA′ , dB′ ). The three mode Gaussian states was also completely solved PPT condition leads to another restriction (Simon, 2003) in terms of operational criterion. 2 2 (∆X(d)) + (∆X(d′)) d~ J d~′ d~ J d~′ , Operational necessary and sufficient condition. In h i̺ h i̺ ≥ | A A A − B B B | (Giedke et al., 2001b) operational necessary and suffi- (267) cient condition for separability for all bipartite Gaus- which may be further written in one combined inequality ~ ~ ~ ~ sian states have been presented. It is so far the only with dAJAd′ + dBJBd′ in its RHS. Special case of this | A| | B | operational criterion of separability that detects all PPT inequality was considered by (Giovannetti et al., 2003) entangled states within such broad class of states. En- together with relation to other criteria. Note that the tanglement is detected via a finite algorithm, that trans- above criterion is only necessary for the state to satisfy forms the initial covariance matrix into a sequence of PPT criterion since it refers to the variance including matrices which after finite number of steps (i) either be- only the first and second moments. comes not physical (not represents a covariance matrix) The practical implementation of PPT criterion and then the algorithm detects entanglement (ii) or its in terms of all moments that goes beyond vari- special affine transformation becomes physical and then ance properties of CV states is Shchukin-Vogel crite- the initial state is recognized to be separable. rion (Miranowicz and Piani, 2006; Shchukin and Vogel, 2005a). It turned out that their criterion covers many known separability criteria. The idea is that with any D. General separability criteria for continuous variables state of two modes, one can associate the following ma- trix of moments One of the natural separability criteria is local pro- q p n m l k r s M = Tr(ˆa† aˆ aˆ† aˆ ˆb† ˆb ˆb† ˆb ρ ), (268) jection or — in general LOCC transformation of CV ij ⊗ AB state onto product of finite dimensional Hilbert spaces where i = (pqrs) and j = (nmkl). The operators a,b and then application of one of separability criteria act on systems A, B respectively. It turns out that the for discrete variables. This method was used in above matrix is positive if and only if the state is PPT79. (Horodecki and Lewenstein, 2000) where finally discrete Positivity of matrix can be expressed in terms of non- variables range criterion was applied. negativity of subdeterminants. It then turns out that First, it must be stressed that any Gaussian separa- many known separability criteria are obtained by impos- bility criterion that refers only to well defined variances, ing nonnegativity of a suitably chosen subdeterminant. and does not use the fact that the variance matrix com- An example is Simon criterion which for 1 1 pletely describes the state, is also a separability criterion Gaussian states is equivalent to PPT. One finds that× for general CV states. it is equivalent to nonnegativity of determinant of a Separability criteria that do not refer to discrete quan- 5 5 main submatrix the matrix M. Other crite- tum states usually are based on some uncertainty type ria× published in (Agarwal and Biswas, 2005; Duan et al., relations. As an example of such relation, consider posi- 2000; Hillery and Zubairy, 2006; Mancini et al., 2002; tion and momenta operators QA1 , QA2 , PA1 , PA2 for a bi- Raymer et al., 2003) can be also reduced to positivity partite system A1A2, which satisfy the commutation re- conditions of some determinants of matrix of moments. 1 lations [QAi , PAj ]= iδij and define U = a QA1 + a QA2 , The approach of Shchukin and Vogel was developed 1 | | V = a PA1 + a PA2 for arbitrary nonzero real number in (Miranowicz et al., 2006). The authors introduced a a. Then| | any separable bipartite CV state ̺ satisfies modified matrix of moments: ′ ′ ′ ′ (Duan et al., 2000) k1 k2 k k l1 l2 l l M ′ ′ = Tr((ˆa† aˆ )†aˆ† 1 aˆ 2 (ˆb† ˆb )†ˆb† 1 ˆb 2 ρ ), kk ll ⊗ AB 1 1 (269) (∆U)2 + (∆V )2 (a2 + ). (265) h i̺ h i̺ ≥ 2 a2 The above criterion can be modified to the form which refers to so called type II standard form of two mode 79 See (Verch and Werner, 2005) in this context. 77 so that it is labeled with four indices, and can be treated monotonous under Gaussian operations. For two-mode itself (up to normalization) as a state of compound sys- Gaussian states its value can be found analytically. If tem. Now, it turns out that the original state is separa- additionally, the state is symmetric with respect to sites, ble, then so is the matrix of moments. In this way, one this measure is additive. On a single copy it is shown to can obtain new separability criteria by applying known be equal to EF . separability criteria to matrix of moments. The PPT The idea of Gaussian entanglement of formation condition applied to matrix of moments turns out to be has been extended to other convex-roof based en- equivalent to the same condition applied to original state. tanglement measures in (Adesso and Illuminati, 2005). Thus applying PPT to matrix of moments reproduces The log-negativity of Gaussian states defined already Shchukin-Vogel result. What is however intriguing, that in (Vidal and Werner, 2002) has also been studied in so far no criterion stronger than PPT was found. (Adesso and Illuminati, 2005). In this case the analytic formula has been found, in terms of symplectic spectrum λi of the partially transposed covariance matrix: E. Distillability and entanglement measures of Gaussian n states E = log [min(1, λ )]. (270) N − 2 i i=1 X The question of distillability of Gaussian states has The continuous variable analogue tangle (squared attracted a lot of effort. In analogy to the two-qubit dis- concurrence, see sec XV), called contangle was intro- tillability of quantum states in finite dimensions, it has duced in (Adesso and Illuminati, 2006) as the Gaussian been first shown, that all two-mode entangled Gaussian convex roof of the squared negativity. It is shown, states are distillable (Giedke et al., 2000). Subsequently that for three-mode Gaussian states contangle exhibits it was shown, that all NPT entangled Gaussian states Coffman-Kundu-Wootters monogamy. Recently the gen- are distillable (Giedke et al., 2001a). In other words, eral monogamy inequality for all N-mode Gaussian states there is no NPT bound entanglement in Gaussian con- was established (Hiroshima et al., 2007) (in full anal- tinuous variables: any NPT Gaussian state can be trans- ogy with the qubit case (Osborne and Verstraete, 2006)). formed into NPT two-mode one, and then distilled as For three modes, the 3-contangle - analogue of Coffman- described in (Giedke et al., 2000). However the proto- Kundu-Wooters 3-tangle is monotone under Gaussian op- col which achieves this task involves operations which erations. are not easy to implement nowadays. The operations Surprisingly, there is a symmetric Gaussian state which feasible for present linear-optic based technology are so is a counterpart of both GHZ as well as W state called Gaussian operations. The natural question was (Adesso and Illuminati, 2006). Namely, in finite dimen- risen then, whether entangled Gaussian states are dis- sion, when maximizing entanglement of subsystems, one tillable by means of this restricted class of operations. obtains W state, while maximization of tangle leads to Unfortunately, it is not the case: one cannot obtain pure GHZ state. For Gaussian states, such optimizations (per- entanglement from Gaussian states using only Gaussian formed for a fixed value of mixedness or of squeezing of operations (Giedke and Cirac, 2002) (see in this context subsystems) leads to a single family of pure states called (Fiurasek, 2002b) and (Eisert et al., 2002a)). Although GHZ/W class. Thus to maximize tripartite entangle- these operations are restrictive enough to effectively ment one has to maximize also bipartite one. “bind” entanglement, they are still useful for process- An exemplary practical use of Gaussian states ing entanglement: by means of them, one can distill key apart from the quantum key distribution (e.g. from entangled NPT Gaussian states (Navascues et al., (Gottesman and Preskill, 2001)) is the application 2004). Interestingly, no PPT Gaussian state from which for continuous quantum Byzantine agreement protocol key can be distilled is known so far (Navascues and Acin, (Neigovzen and Sanpera, 2005). There are many other 2005). theoretical and experimental issues concerning Gaussian Apart from question of distillability and key distill- states and their entanglement properties, that we do ability of Gaussian states, entanglement measures such not touch here. For a recent review on this topic see as entanglement of formation and negativity have been (Adesso et al., 2007; Ferraro et al., 2005). studied. It led also to new measures of entanglement called Gaussian entanglement measures. In (Giedke et al., 2003b) entanglement of formation XVIII. MISCELLANEA was calculated for symmetric Gaussian states. Interest- ingly, the optimal ensemble realizing EF consists solely A. Entanglement under information loss: locking from Gaussian states. It is not known to hold in gen- entanglement eral. One can however consider the so called Gaussian entanglement of formation EG where infimum is taken Manipulating a quantum state with local operations over decompositions into Gaussian states only. Gaussian and classical communication in non-unitary way usually entanglement of formation was introduced and studied decreases its entanglement content. Given a quantum bi- in (Giedke et al., 2003b). It is shown there, that E is partite system of 2 log d qubits in state ρ one can ask G × 78 how much entanglement can decrease if one traces out The examples of states for which EF can change from a single qubit. Surprisingly, a lot of entanglement mea- nearly maximal (log d) to nearly zero after tracing out sures can decrease by arbitrary large amount, i.e. from O(log log d) qubits was found in (Hayden et al., 2006). O(log d) to zero. Generally, if some quantity of ρ can It can indicate either drastic nonadditivity of entangle- decrease by an arbitrarily large amount (as a function of ment of formation, or most probably an extreme case of number of qubits) after LOCC operation on few qubits, irreversibility in creation-distillation LOCC processes81. then it is called lockable. This is because a huge amount The analogous effect has been found earlier in classical of quantity can be controlled by a person who posses only key agreement (Renner and Wolf, 2003). It was shown a small dimensional system which plays a role of a “key” that intrinsic information I(A : B Ee) (an upper to this quantity. bound on secret key extractable from↓ triples of random The following related question was asked earlier in variables) can decrease arbitrarily after erasing 1-bit ran- (Eisert et al., 2000a): how entanglement behaves under dom variable e (see Sec. XIX.F). classical information loss? It was quantified by means of In (Christandl and Winter, 2005) it was shown that entropies and for convex entanglement measures it takes squashed entanglement is also lockable measure, which is the form again exhibited by flower states. The authors has shown also that regularized entanglement of purification E is ∆E ∆S (271) p∞ ≤ lockable quantity, as it can go down from maximal to where ∆E = i piE(ρi) E( i piρi and zero after erasing one-qubit system a of the state ∆S = S(ρ) p s(ρ ). It holds− for relative en- − i i i tropy of entanglementP (Linden et al., 1999a)P (see also d +1 (+) d 1 ( ) ω = 0 0 P + − 1 1 P − , (273) (Synak-Radtke andP Horodecki, 2006)). It turns out aAB 2d | ih |a⊗ AB 2d | ih |a⊗ AB however that this inequality can be drastically violated, 80 ( ) due to above locking effect . with PAB± being normalized projectors onto symmet- The phenomenon of drastic change of the content of ric and antisymmetric subspace respectively (see Sec. state after tracing out one qubit, was earlier recognized VI.B.3). Although Ep∞ is not an entanglement measure, in (DiVincenzo et al., 2004) in the case of classical cor- it is apart from EC another example of lockable quan- relations of quantum states (maximal mutual informa- tity which has operational meaning (Terhal et al., 2002). tion of outcomes of local measurements) (see in this con- In case of (273) Ep = Ep∞, so the Ep itself also can be text (Ballester and Wehner, 2006; Buhrman et al., 2006; locked. Koenig et al., 2005; Smolin and Oppenheim, 2006)). An- It is known that all measures based on the convex other effect of this sort was found in a classical key agree- roof method (see Sec. XV.C.2) are lockable. There ment (Renner and Wolf, 2003) (a theory bearing some is also a general connection between lockability and analogy to entanglement theory, see sec. XIX.F). asymptotic non-continuity. Namely it was shown in Various entanglement measures have been shown to be (Horodecki et al., 2005d) that any measure which is (i) lockable. In (Horodecki et al., 2005d) it has been shown subextensive i.e. bounded by M log d for constant M, that entanglement cost, and log-negativity are lockable (ii) convex (iii) not asymptotically continuous, is lock- measures. The family of states which reveal this property able. The only not lockable measure known up to date of measures (called flower states) can be obtained via is the relative entropy of entanglement (Horodecki et al., partial trace over system E of the pure state 2005d), which can be derived with help of (271) d 1 Since fragility of measure to one qubit erasure is a cu- 1 − ψ(d) = i 0 i 0 i (272) rious property, for any lockable entanglement measure E | AaBEi √ | iA| ia{| i| i}B| iE 2d i=0 one could design its reduced version, that is a non-lockable X version of E. One of the possible definitions of reduced + i A 1 a i 1 BU i E, | i | i {| i| i} | i entanglement measure is proposed in (Horodecki et al., log d where U = H⊗ is a tensor product of Hadamard 2005d). Possible consequences of locking for multipartite 1 transformations. In this case EC (ρAaB) = 2 log d and entanglement measures can be found in (Groisman et al., EN (ρAaB ) = log(√d + 1). However after tracing out 2005). qubit a one has EC (ρAB ) = EN (ρAB) = 0, as the state Although the phenomenon of locking shows that cer- is separable. The gap between initial EC and final one, tain measures can highly depend on the structure of par- can be made even more large, using more unitaries in- ticular states, it is not known how many states have a stead of U and I in the above states and the ideas of structure which can contribute to locking of measures, randomization (Hayden et al., 2004). and what kind of structures are specific for this purpose. 80 Using the fact that that a loss of one qubit can be simulated 81 Because distillable entanglement of the states under considera- by applying one of four random Pauli matrices to the qubit one tion is shown to be small we have either EF >> EC (reporting easily arrives at the connection between locking, and violation of nonadditivity of EF ) or EF = EC >> ED (reporting extreme the above inequality. creation-distillation irreversibility). 79 It is an open question whether distillable entanglement destroyed, so that final entanglement is zero. can be locked, although it is known that its one-way ver- This approach was developed in (Horodecki et al., sion is lockable, which follows from monogamy of entan- 2003c) (in particular, it was shown that any measure glement. In case of distillable key, one can consider two of entanglement can be used) and it was proved that a versions of locking: the one after tracing out a qubit full basis cannot be distinguished even probabilistically, from Eve (E-locking), and the one after the qubit of Al- if at least one of states is entangled. Other results can ice’s and Bob’s system is traced out (AB-locking). It be found in (Fan, 2004; Nathanson, 2005; Virmani et al., has been shown, that both classical and quantum dis- 2001). In (Virmani et al., 2001) it was shown that for tillable key is not E-lockable (Christandl et al., 2006; any two pure states the famous Helstrom formula for Renner and Wolf, 2003). It is however not known if dis- maximal probability of distinguishing by global measure- tillable key can be AB lockable i.e. that if erasing a ments, holds also for LOCC distinguishing. qubit from Alice and Bob− systems may diminish by far In (Badzi¸ag et al., 2003) a general quantitative bound their ability to obtain secure correlations. Analogous for LOCC distinguishability was given in terms of the question remains open in classical key agreement (see Sec. maximal mutual information achievable by LOCC mea- 82 LOCC XIX.F). surements Iacc . The bound can be viewed is a gen- eralization of Holevo bound: ILOCC log N E, (274) B. Entanglement and distinguishing states by LOCC acc ≤ − where N is number of states, and E is average en- Early fundamental results in distinguishing by LOCC tanglement (it holds for any E which is convex and is are the following: there exist sets of orthogonal product equal to entropy of subsystems for pure states). The states that are not perfectly distinguishable by LOCC usual Holevo bound would read just Iacc log N, so (Bennett et al., 1999a) (see also (Walgate and Hardy, we have correction coming from entanglement.≤ The fol- 2002)) and every two orthogonal states (even multipar- lowing generalization of the above inequality obtained in tite) are distinguishable by LOCC (Walgate et al., 2000). (Horodecki et al., 2004) In a qualitative way, entanglement was used in the prob- LOCC lem of distinguishability in (Terhal et al., 2001). To show Iacc log N Ein Eout (275) that a given set of states cannot be distinguished by ≤ − − LOCC, they considered all measurements capable to dis- shows that if some entanglement, denoted here by Eout, tinguish them. Then applied the measurements to AB is to survive the process of distinguishing (which is re- part of the system in state ψAA′ ψBB′ where the com- quired e.g in process of distillation) then it reduces dis- 83 ponents are maximally entangled⊗ states. If the state after tinguishability even more . measurement is entangled across AA′ : BB′ cut, then one Finally in (Hayashi et al., 2006) the number M of or- concludes that the measurement cannot be done by use thogonal multipartite states distinguishable by LOCC of LOCC because the state was initially product across was bounded in terms of several measures of entangle- this cut. ment, in particular we have An interesting twist was given in (Ghosh et al., 2001) 1 P S(ρi)+ER(ρi) where distillable entanglement was used. Let us see M D2− M i , (276) ≤ how they argued that four Bell states ψi (3) cannot be distinguished. Consider fourpartite state ρABA′B′ = where D denotes dimension of the total system, S de- 1 ′ ′ notes the von Neumann entropy and ER is the relative 4 i ψi ψi AB ψi ψi A B . Suppose that it is possi- ble to| distinguishih | ⊗ Bell | ih states| by LOCC. Then Alice and entropy of entanglement (see Sec. XV.C.1). A strik- BobP will distinguish Bell states of system AB (perhaps ing phenomenon is possibility of “hiding” discovered in destroying them). Then they will know which of Bell (DiVincenzo et al., 2002; Terhal et al., 2001). Namely, states they share on system A′B′, obtaining then 1 e-bit one can encode one bit into two states ρ0 and ρ1, for of pure entanglement (hence ED 1). However, one can which probability of error in LOCC-distinguishing can ′≥′ be arbitrary close to 1. In (Eggeling and Werner, 2002) check that the initial state ρABA B is separable (Smolin, 2005), so that ED = 0 and we get a contradiction. This it was shown that this can be achieved even with ρi be- shows that entanglement measures can be used to prove ing separable states. Thus unlike for pure states, for two impossibility of distinguishing of some states. Initially mixed states, Helstrom formula does not hold any longer. entanglement measures have not been used for this prob- lem. It is quite nontrivial concept, as naively it could seem that entanglement measures are here useless. In- 82 LOCC It is defined as follows: Iacc ( pi, ψi ) = sup A I(i : j) deed, the usual argument exploiting the measures is the { } { j } following: a given task cannot be achieved, because some where supremum is taken over measurements that can be im- plemented by LOCC, and I(i : j) is mutual information between entanglement measure would increase. In the problem symbols i and measurement outcomes j. of distinguishing, this argument cannot be directly ap- 83 The inequality was independently conjectured and proven for one plied, because while distinguishing, the state is usually way LOCC in (Ghosh et al., 2004) 80 LOCC Finally, a lower bound on Iacc was provided in 2003b) on the basis of purely information-theoretical con- (Sen(De) et al., 2006) and relation with entanglement siderations. It was then also related to thermodynam- distillation was discussed. ics (Zurek, 2000).) Recently other measures of quan- tumness correlations based on distance from classically correlated states were introduced (Groisman et al., 2007; C. Entanglement and thermodynamical work SaiToh et al., 2007). In (Oppenheim et al., 2002a) an approach to com- Localisable information is a notion dual to distillable posite systems was proposed based on thermodynamics. entanglement: instead of singlets we want to distill prod- Namely, it is known, that given a pure qubit and a heat uct states. Thus local states constitute valuable resource bath, one can draw amount kT ln2 of work84. The second (the class of involved operations exclude bringing fresh law is not violated, because afterward the qubit becomes local ancillas). It turns out that techniques from en- mixed serving as an entropy sink (in place of reservoir of tanglement distillation can be applied to this paradigm lower temperature) (see (Alicki et al., 2004; Scully, 2001; (Horodecki et al., 2003a; Synak et al., 2005). One of the Vedral, 1999)). In this way the noisy energy from the main connections between distillation of local purity and heat bath is divided into noise, that goes to the qubit, entanglement theory is the following: For pure bipartite and pure energy that is extracted. More generally, a d states the quantum deficit is equal to entropy of entan- level system of entropy S allows to draw amount of work glement (Horodecki et al., 2003a). Thus one arrives at equal to kT (log d S). If we treat I = log d S as in- entanglement by an opposite approach to the usual one: formation about the− state (it is also called purity),− then instead of looking at nonlocal part, one considers local the work is proportional to information/purity (Devetak, information, and subtracts it from total information of 2004). the state. It is an open question, if for pure multipartite Suppose now that Alice and Bob have local heat baths, states, the quantum deficit is also equal to some entan- and want to draw work from them by means of shared bi- glement measure (if so, the measure is most likely the partite state. If they do not communicate, then they can relative entropy of entanglement). only draw the work proportional to local purity. If they Another link is that the above approach allows to de- share two maximally classically correlated qubits , local fine cost of erasure of entanglement in thermodynamical states are maximally mixed, and no work can be drawn. spirit. Namely the cost is equal to entropy that has to However if Bob will measure his qubit, and tell Alice the be produced while resetting entangled state to separable value, she can conditionally rotate her qubit, so that she one, by closed local operations and classical communica- obtains pure qubit, and can draw work. For classically tion (Horodecki et al., 2005g). (closed means that we do correlated systems, it turns out that the full informa- not trace out systems, to count how much entropy was tion content log d S can be changed into work. How produced). It was proved that the cost is lower bounded much work can Alice− and Bob draw from a given quan- by relative entropy of entanglement. (some related ideas tum state, provided that they can communicate via a have been heuristically claimed earlier in (Vedral, 1999)). classical channel? It turns out that unlike in the classical case, part of information cannot be changed into work, Note that another way of counting cost of entangle- because it cannot be concentrated via classical channel. ment erasure is given by robustness of entanglement. In For example, for singlet state the information content is that case it is a mathematical function, while here the 2, but only 1 bit of work (in units kT) can be drawn. cost is more operational, as it is connected with entropy The amount of work that can be drawn in the above production. paradigm is proportional to the amount of pure lo- cal qubits that can be obtained (localisable informa- In a recent development the idea of drawing work from tion/purity). The difference between total informa- local heat baths have been connected with a separability tion I and the localisable information represents purely criterion. Namely, in (Maruyama et al., 2005) the follow- quantum information, and is called quantum deficit ing non-optimal scheme of drawing work was considered (Oppenheim et al., 2002a). It can be viewed as a mea- for two qubits: Alice and Bob measure the state in the sure of quantumness of correlations, which is zero for a same randomly chosen basis. If they share singlet state 85 classically correlated states . (A closely related quantity ψ−, irrespectively of choice of basis, they obtain perfect called quantum discord was obtained earlier in (Zurek, anti-correlations, allowing to draw on average 1 bit (i.e. kT ln2) of work. Consider instead classically correlated 1 state 2 ( 00 00 + 11 11 ). We see that any measure- ment apart| ih from| the| ih one| in bases 0 , 1 will decrease 84 It is reverse of Landauer principle, which says that resetting correlations. Thus if we average over| choicei | i of bases, the bit/qubit to pure state costs the same amount of work. drawn work will be significantly smaller. The authors 85 A bipartite state is properly classically correlated or shortly have calculated the optimal work that can be drawn from classically correlated, if it can be written in the form: ρAB = P pij i i j j with coefficients 0 pij 1, P pij = 1; separable states, so that any state which allows to draw ij | ih |⊗| ih | ≤ ≤ ij i and j are local bases (Oppenheim et al., 2002a). more work, must be entangled. {| i} {| i} 81 D. Asymmetry of entanglement gate capacities (Harrow and Shor, 2005; Linden et al., 2005), and the possibility of exchange of the subsys- To characterize entanglement one should recognize its tems of a bipartite state in such a way that entan- different qualities. Exemplary qualities are just transi- glement with a purifying reference system is preserved tion rates (exact or asymptotic) between different states (Oppenheim and Winter, 2003). Although these are very under classes of operations such as LOCC (see Secs. different phenomena, the possible relation is not ex- XIII.A and XIII.B). cluded. Recently, in (Horodecki et al., 2005a) a different type of transition was studied. Namely, transition of a bipar- tite state into its own swapped version: ρ V ρV with XIX. ENTANGLEMENT AND SECURE CORRELATIONS V being a swap operation which exchanges−→ subsystems of ρ (see Sec. (48)). In other words this is transition A fundamental difference between classical and quan- from ρAB into ρBA under LOCC. If such a transition tum, is that quantum formalism allows for states of com- is possible for a given state, one can say, that entangle- posite systems to be both pure and correlated. While in ment contents of this state is symmetric. Otherwise it is classical world those two features never meet in one state, asymmetric, and the asymmetry of entanglement can be entangled states can exhibit them at the same time. quantified as follows For this reason, entanglement in an astonishing way A(ρ) = sup Λ(ρ) V ρV 1, (277) incorporates basic ingredients of theory of secure com- Λ k − k munication. Indeed, to achieve the latter, the interested where supremum is taken over all LOCC operations Λ. persons (Alice and Bob) need a private key: a string of Note that if a state is symmetric under swap, then also bits which is i) perfectly correlated (correlations) and ii) its entanglement contents is symmetric, however the con- unknown to any other person (security or privacy) (then verse does not hold (a trivial example is a pure nonsym- they can use it to preform private conversation by use metric state: one can produce its swapped version by of so-called Vernam cipher (Vernam, 1926) ). Now, it local unitaries). is purity which enforces the second condition, because It was shown that there exist states entanglement of an eavesdropper who wants to gain knowledge about a which is asymmetric in the above sense. Namely, one can quantum system, will unavoidably disturb it, randomiz- prove if the entanglement measure called G-concurrence ing phase via quantum back reaction. In modern ter- (Fan et al., 2003; Gour, 2005; Sino l¸ecka et al., 2002) is minology we would say, that if Eve applies CNOT gate, nonzero, then the only LOCC operation that could swap to gain knowledge about a bit, at the same time she in- the state is just local unitary operation. Thus any state troduces a phase error into the system, which destroys which has nonzero G-concurrence, and has different spec- purity, see (Zurek, 1981). tra of subsystems cannot be swapped by LOCC and Let us note that all what we have said can be phrased hence any it contains asymmetric entanglement. It is an in terms of monogamy of entanglement in its strong open question if there exist states which maintain this version: If two quantum systems are maximally quan- asymmetry even in asymptotic limit of many copies. tumly correlated, then they are not correlated with any One should note here, that the asymmetry in con- other system at all (neither quantumly nor classically) text of quantum states has been already noted in lit- (see (Koashi and Winter, 2004) in this context). In this erature. In particular the asymmetric correlation func- section we shall explore the mutual interaction between tions f of a quantum state are known, such as e.g. one- entanglement theory and the concept of private correla- tions. way distillable entanglement ED→ (Bennett et al., 1996a) or quantum discord (Zurek, 2000). Considering properly symmetrised difference between f(ρAB) and f(ρBA) one could in principle obtain some other measure of asymme- A. Quantum key distribution schemes and security proofs try of quantum correlations. However it seems that such based on distillation of pure entanglement a measure would not capture the asymmetry of entangle- ment content exactly. The functions defined by means of Interestingly, the first protocol to obtain a private key 86 asymmetric (one-way) class of operations like ED→ seems - the famous BB84 (Bennett and Brassard, 1984), did to bring in a kind of its ”own asymmetry” (the same holds not use the concept of entanglement at all. Neither e.g. for one-way distillable key). Concerning the corre- uses entanglement another protocol B92 proposed by Ch. lation functions f such as quantum discord, they seem Bennett in 1992 (Bennett, 1992), and many variations of to capture the asymmetry of quantum correlations in BB84 like six-state protocol (Bruß, 1998). Indeed, these general rather then that of entanglement, being nonzero QKD protocols are based on sending randomly chosen for separable states (the same holds e.g. for Henderson- nonorthogonal quantum states. Alice prepares a random Vedral CHV quantity (see 222) and one-way distillable common randomness (Devetak and Winter, 2004a)). The phenomenon of asymmetry has been studied also in other contexts, such as asymmetry of quantum 86 We call such protocol Quantum Key Distribution (QKD). 82 signal state, measures it and send it to Bob who also protocols of entanglement distillation (fundamental for measures it immediately after reception. Such protocols the whole quantum communication theory) have been are called a prepare and measure protocols (P& M). designed by use of methods of generation of secure key The first entanglement based protocol was discovered (Bennett et al., 1996c,d). by Ekert (see Sec. III). Interestingly, even Ekert’s pro- Another (at least theoretical) advantage of tocol though using explicitly entanglement was still not entanglement-based QKD protocol is that with quan- based solely on the “purity & correlations” concept out- tum memory at disposal, one can apply dense coding lined above. He did exploit correlations, but along with scheme and obtain a protocol which has higher capacity purity argument, used violation of Bell inequalities. It than usual QKD as it was applied by Long and Liu seems that it was the paper of Bennett, Brassard and (Long and Liu, 2002) (see also (Cabello, 2000)). Mermin (BBM) (Bennett et al., 1992) which tilted the Moreover, entanglement may help to carry out quan- later history of entanglement-based QKD from the “Bell tum cryptography over long distances by use quantum re- inequalities” direction to “disturbance of entanglement”: peaters which exploit entanglement swapping and quan- upon attack of Eve, the initially pure entangled state tum memory (D¨ur et al., 1999a). We should note that all becomes mixed, and this can be detected by Alice and the above potential advantages of entanglement, would Bob. Namely they have proposed the following protocol need quantum memory. which is also entanglement based, but is not based on Bell inequality. Simply Alice and Bob, when given some (untrusted) EPR pairs check their quality by measuring 1. Entanglement distillation based quantum key distribution correlations in 0 , 1 basis, and the conjugated basis protocols. = 1 ( 0 {| 1i ).| i} So simplified Ekert’s protocol is |±i √2 | i ± | i Both Ekert’s protocol, and its BBM version worked in formally equivalent to the BB84 protocol. Namely the situation, where the disturbance comes only from eaves- total final state between Alice, Bob and Eve is the same dropper, so if only Alice and Bob detect his presence, in both cases87. Thus entanglement looks here quite su- they can abort the protocol. Since in reality one usually perfluous, and moreover Bell inequalities appear rather deals with imperfect sources, hence also with imperfect accidentally: just as indicators of possible disturbance by (noisy) entanglement, it is important to ask if the secure Eve. key can be drawn from noisy EPR pairs. The purifi- Paradoxically, it turned out recently that the Bell in- cation of EPR pairs appeared to be crucial idea in this equalities are themselves a good resource for key distribu- case. The first scheme of purification (or distillation) tion, and allow to prove security of private key without of entanglement has been discovered and developed in assuming quantum formalism but basing solely on no- (Bennett et al., 1996c,d) (see Sec. XII). In this scheme signaling assumption. (Acin et al., 2006a; Barrett et al., Alice and Bob share n copies of some mixed state, and 2005; Masanes and Winter, 2006). There is still an anal- by means of local quantum operations and classical com- ogy with entanglement: nonlocal correlations are monog- munication (LOCC) they obtain a smaller amount k This problem has been tackled by Lo and Chau Bell states which are all correlated (Lo and Chau, 1999) who have provided the first both 1 unconditionally secure and fully entanglement-based φ− = ( 00 11 ), scheme89. To cope with imperfections Alice and Bob | i √2 | i − | i use fault tolerant quantum computing. In order to obtain 1 φ+ = ( 00 + 11 ). (280) secure key, they perform entanglement distillation proto- | i √2 | i | i col of (Bennett et al., 1996d) to distill singlets, and check their quality90. Then they apply phase error correcting procedure. I.e. they get to know which systems are in φ− and which in The Lo-Chau proposal has a drawback: one needs a + + φ so that they can rotate each φ− into φ and finally quantum computer to implement it. On the other hand, + the first quantum cryptographic protocol (BB84) does obtain a sequence of φ solely. What Shor and Preskill noticed is that these two quantum procedures are co- not need a quantum computer. And the BB84 was al- 91 ready proved to be secure by Mayers (Mayers, 2001). Yet, herent versions of classical error-correction and privacy the proof was quite complicated, and therefore not easy amplification respectively. Thus Shor-Preskill proof can to generalize to other protocols. be phrased as follows: “The BB84 protocol is secure be- cause its suitable coherent version distills EPR states”. It should be emphasized here, that the equivalence be- tween noisy BB84 protocol and its coherent version does not continue to the very end. Namely, in distillation, Al- 2. Entanglement based security proofs ice and Bob after finding which pair is in φ− and which + | i in φ , rotate φ− . The classical procedures performed | i | i A remarkable step was done by Shor and Preskill coherently cannot perform this very last step (rotation) (Shor and Preskill, 2000), who showed, that one can as no classical action can act as phase gate after embed- prove security of BB84 scheme, which is a P&M proto- ding into quantum. However, the key is secure, because col, by considering mentally a protocol based on entan- Alice and Bob could have performed the rotation, but glement (a modified Lo-Chau protocol). This was some- they do not have to. Indeed, note that if Alice and Bob thing like the Bennett, Brassard, Mermin consideration, measure the pairs in basis 0 , 1 they obtain the same {| i | i} but in a noisy scenario. Namely, while using BB84 in results, independently of whether they have rotated φ− | i presence of noise, Alice and Bob first obtain a so-called or not. The very possibility of rotation, means that the raw key - a string of bits which is not perfectly correlated key will be secure. Thus the coherent version of BB84 (there are some errors), and also not perfectly secure (Eve does not actually give φ+ itself, but it does if supple- | i have some knowledge about the key). By looking at the mented with rotations. part of the raw key, they can estimate the level of error The concept of proving security of P&M protocols by and the knowledge of Eve. They then classically process showing that at the logical level they are equivalent to it, applying procedures of error correction and privacy distillation of entanglement, has become very fruitful. In amplification (the latter aims at diminishing knowledge 2003 K. Tamaki, M. Koashi and N. Imoto (Tamaki et al., of Eve). 2003) showed that B92 is unconditionally secure, using In related entanglement based scheme, we have co- Shor-Preskill method (see also (Tamaki and L¨utkenhaus, herent analogues of those procedures. Without going 2004)). They showed that B92 is equivalent to special en- into details, we can imagine that in entanglement based tanglement distillation protocol known as filtering (Gisin, scheme, Alice and Bob share pairs in one of four Bell 1996b; Horodecki et al., 1997) (see section XII.D). In states (3), which may be seen as the state φ+ with two (Ardehali et al., 1998) the efficient version of BB84 was kinds of errors: bit error, and phase error. The error proposed, which is still unconditionally secure though the from the previous scheme translates here into bit error, while knowledge of Eve’s into phase error. Now the task is simply to correct both errors. Two procedures of a dif- 91 ferent kind (error correction and privacy amplification) It is worth to observe that formally any QKD scheme can be made “entanglement based”, as according to axioms of quantum are now both of the same type - they correct errors. mechanics any operation is unitary and the only source of ran- After correcting bit error, Alice and Bob are left with domness is the subsystem of an entangled state. From this point of view, even when Alice sends to Bob a randomly chosen signal state, as it is in P&M schemes, according to axioms she sends a part of entangled system. Moreover, any operation that Alice and Bob would perform on signal states in P&M scheme, can be 89 The first proof of unconditional security of quantum key distri- done reversibly, so that the whole system shared by Alice Bob bution was provided by Mayers who proved security of BB84 and Eve is in a pure state at each step of the protocol. This (Mayers, 2001). principle of maintaining purity is usually referred to as coherent 90 Using the concept of another entanglement-based communication processing. Let us note however, that not always the coherent scheme — quantum repeaters (Briegel et al., 1998; D¨ur et al., application of a protocol that provides a key, must result in the 1999a) Lo and Chau established quantum key distribution distillation of φ+. In Sec. XIX.B.2 we will see that there is a (QKD) over arbitrary long distances. more general class of states that gives private key. 84 number of systems that Alice and Bob use to estimate the 2006a,b). It is also connected with optimization of en- error rate is much smaller than in BB84. Again security tanglement measures from incomplete experimental data is proved in Shor-Preskill style. In (Gottesman and Lo, (see Sec. VI.B.4). 2003) a P&M scheme with a two-way classical error cor- rection and privacy amplification protocol was found. It is shown that a protocol with two-way classical com- 4. Secure key beyond distillability - prelude munication can have substantially higher key rate than the one with the use of one-way classical communica- The fact that up to date techniques to prove uncondi- tion only. Also security of key distribution using dense tional security were based on entanglement purification coding (Long and Liu, 2002) was proved by use of Shor- i.e. distilling pure entangled states, has supported the be- Preskill techniques (Zhang et al., 2005) (see in this con- lief that possibility of distilling pure entanglement (sin- text (Degiovanni et al., 2003, 2004; W´ojcik, 2005)). glet) is the only reason for unconditional security. For Thanks to simplicity of Shor-Preskill approach pure this reason the states which are entangled but not distil- entanglement remained the best tool for proving uncon- lable (i.e. bound entangled) found in (Horodecki et al., ditional security of QKD protocols (Gisin and Brunner, 1998a) have been considered as unlikely to be useful 2003). As we will see further this approach can be gener- for cryptographical tasks. Moreover Acin, Masanes and alized by considering mixed entangled states containing Gisin (Acin et al., 2003b), showed that in the case of two ideal key. qubit states (and under individual Eve’s attack) one can distill key by single measurement and classical postpro- cessing if and only if the state contains distillable entan- 3. Constraints for security from entanglement glement. Surprisingly, as we will see further, it has been recently proved that one can obtain unconditionally se- So far we have discussed the role of entanglement in cure key even from bound entangled states. particular protocols of quantum key distribution. A The first interesting step towards this direction was connection between entanglement and any QKD pro- due to Aschauer and Briegel, who showed Lo and tocol has been established by Curty, Lewenstein and Chau’s protocol provide key even without the fault tol- L¨utkenhaus (Curty et al., 2004). They have proved that erant computing i.e. with realistic noisy apparatuses entanglement is necessary precondition of unconditional (Aschauer and Briegel, 2002). security. Namely, in case of any QKD protocol Alice The crucial property of their approach was to consider and Bob perform some measurements and are left with all noise, (which they assumed to act locally in Alice and some classical data from which they want to obtain key. Bob’s laboratories) in a coherent way. Keeping track of Basing on these data and measurements settings, they any noise introduced by imperfect devices, they can en- must be able to construct a so called entanglement wit- sure that the total state of distilled noisy singlets together ness to ensure that the data could not be generated via with the state of their laboratories is in fact a pure state. measurement on some separable state (see Sec. VI.B.3 ). This holds despite the fact that the noisy singlets may Let us emphasized, that this holds not only for entangle- have fidelity, that is overlap with a state (6), bounded ment based but also for prepare and measure protocols. away from maximal value 1. This kind of noisy entan- In the latter case, a kind of “effective” entanglement is glement, which is corrupted because of local noise, but witnessed (i.e. the one which is actually never shared by not due to the eavesdropping, they have called private Alice and Bob). entanglement. It is worth noting, that this approach can be seen as a generalization of the very first Ekert’s approach (Ekert, 1991). This is because Bell inequalities can be seen as a B. Drawing private key from distillable and bound n special case of entanglement witnesses (see Sec. VI.B.5). entangled states of the form ρ⊗ Formally, in any QKD protocol, Alice and Bob perform repeatedly some POVM’s Ai and Bi respectively, Strong interrelation between theory of secure key and { } { } and obtain a probability distribution of the outcomes entanglement can already be seen in the scenario, where P (a,b). Now, a necessary condition for security of the Alice and Bob share n bipartite systems in the same protocol is that it is possible to build out some entangle- state ρAB and Eve holds their purification, so that the ment witness W = cij Ai Bj with real cij , such joint state of Alice, Bob and Eve systems is a pure state i,j ⊗ that c p < 0 and TrW σ 0 for all separable ψABE . The task of the honest parties is to obtain by ij ij ij | i states σ. Indeed, otherwiseP they≥ could make key from means of Local Operations and Classical Communication separableP states which is impossible (Gisin and Wolf, (LOCC) the highest possible amount of correlated bits, 2000) (see also discussion in sec XIX.B). This idea that are unknown to Eve (i.e. a secure key). The diffi- has been studied in case of high dimensional sys- culty of this task is due to the fact that Eve makes a copy tems (Nikolopoulos and Alber, 2005; Nikolopoulos et al., of any classical message exchanged by Alice and Bob. 2006) and general upper bounds on key rates for prepare The above paradigm, allows to consider a new measure and measure schemes has been found (Moroder et al., of entanglement: distillable key KD, which is similar in 85 spirit to distillable entanglement as it was discussed in where I(X : Y ) = S(X)+ S(Y ) S(XY ) is quantum Secs. XV.D.1 and XV.A. It is given by the number of mutual information. − secure bits of key that can be obtained (per input pair) Without going into details, let us note that the quan- from a given state. tity I(A : B) is the common information between Alice Let us discuss in short two extreme cases: and Bob, hence it says how many correlated bits Alice and Bob will obtain via error correction. Since I(A : E) All distillable states are key distillable: • is common information between Eve and Alice, its sub- KD(ρAB ) ED(ρAB). (281) traction means, that in the procedure of privacy ampli- ≥ fication this amount of bits has to be removed from key, All separable states are key non-distillable: to obtain a smaller key, about which Eve does not know • anything. KD(σsep)=0. (282) Now, let us note that in the present case we have To see the first statement, one applies the idea of quan- I(A : B)= S(ρ ) p S(ρi ), tum privacy amplification described above. Simply, Alice B − i B i and Bob distill singlets and measure them locally. Due X i to “purity & correlation” principle, this gives a secure I(A : E)= S(ρE) piS(ρ ), (285) − E i key. X To see that key cannot be drawn from separable states i i i i B (Curty et al., 2004; Gisin and Wolf, 2000), note that by where ρB = TrEρBE, ρE = TrBρBE, ρB = i piρi = E definition of separability, there is a measurement on Eve’s TrAρAB, and ρE = i piρi = TrABρAE. P subsystem such that conditionally upon result (say i) Al- Since the measurement of Alice was complete, the i P i i (i) (i) states ρBE are pure, hence S(ρE) = S(ρB). Also, since ice and Bob share a product state ρA ρB . This means that Alice and Bob conditionally on Eve⊗ have initially no the total initial state was pure, we have S(ρE)= S(ρAB) correlations. Of course, any further communication be- where ρAB is initial state shared by Alice and Bob. Thus tween Alice and Bob cannot help, because it is monitored we obtain that the amount of key gained in the protocol, by Eve. is actually equal to coherent information. In Sec. XIX.B.6 it will be shown, that there holds: Now, Devetak and Winter have applied the protocol coherently, and obtained protocol of distillation of sin- There are non-distillable states which are key dis- glets proving first general lower bound for distillable en- • tillable: tanglement ED(ρbe)=0 & KD(ρbe) > 0. (283) E (ρ ) S(ρ ) S(ρ ). (286) D AB ≥ B − AB This was also used by Devetak to provide the first fully 1. Drawing key from distillable states: Devetak-Winter protocol rigorous proof of quantum Shannon theorem, saying that capacity of quantum channel is given by coherent infor- Here we will present a protocol due to Devetak and mation (Devetak, 2003). Thus cryptography was used Winter, which shows that from any state, one can also to develop entanglement and quantum communica- draw at least amount of key equal to coherent informa- tion theory. tion. This is compatible with the idea, that one can draw key only from entangled states (states with pos- itive coherent information are entangled, as shown in (Horodecki and Horodecki, 1994)). The coherent version 2. Private states of the protocol will in turn distill this amount of singlets from the state. In this way Devetak and Winter have In the previous section we have invoked a protocol, for the first time proved the hashing inequality (XV.F.1) which produces a private key, and if applied coherently, for distillable entanglement. Thus, again cryptographic distills singlets. A fundamental question which naturally techniques allowed to develop entanglement theory. arises is how far this correspondence goes. In particular, one can ask whether it is possible to distill key only from Namely, consider a state ρAB, so that the total state states from which singlets can be distilled. Surprisingly, including Eve’s system is ψABE . As said, we assume that they have n copies of such state. Now Alice performs it has been shown that one can obtain key even from cer- complete measurement, which turns the total state into tain bound entangled states (Horodecki et al., 2005b,e). i To this end they have characterized class of states which ρcqq = i pi i i A ρBE where subscript cqq reminds that Alice’s system| ih | is⊗ classically correlated with Bob and have subsystem that measured in certain product basis Eve subsystems.P The authors considered drawing key gives log d bits of perfect key. The present section is from a general cqq state as a starting point, and showed mostly devoted to this class of states which apart from that one can draw at least the amount of key equal to being fundamental for analysis of key distillation consti- tute a new quality in mixed state entanglement, hence it I(A : B) I(A : E), (284) is also of its own interest. − 86 These states, called private states or gamma states, One can show that in general KD EC , so that have been proved to be of the following form92 E (ρ ) 1. ≤ C γ ≥ d 1 Although twisting can diminish distillable entan- (d) 1 − glement, it can not set it to zero. Indeed it γ = ii jj U ρ ′ ′ U †, (287) d | ih |AB ⊗ i A B j was shown that all private states are distillable i,j=0 X (Horodecki and Augusiak, 2006). + where Ui are arbitrary unitary transformations acting on Interestingly, private state can be written as Φ , | i the system A′B′. The whole state resides on two systems where amplitudes have been replaced by q-numbers: with distinguished subsystems AA′ and BB′ respectively. The AB subsystem, after measuring in computational γ = ΨΨ†, (292) basis, gives log d bits of key, hence it is called the key part of the private state. A private state with d d- where Ψ is the matrix (not necessary a state) similar to dimensional key part is called shortly a pdit and in× spe- a pure state, cial case of d =2a pbit. The subsystem A′B′ aims to d 1 protect the key part, hence it is referred to as a shield − ′ ′ 93 Ψ= Y A B ii , (293) . The class of private state has been generalized to the i ⊗ | iAB i=0 multipartite case (Horodecki and Augusiak, 2006). X ρ with Yi = Ui d for some unitary transformation Ui 3. Private states versus singlets and certain state ρ on A′B′. Looking at the state d 1 1p ψ = − eiφj jj we see that Y is a noncommu- j=0 √d i | i |iφ i A private state can be obtained via unitary transfor- tative version of 1 j . This special form has not yet mation from a simple private state called basic pdit P √d been studied thoughtfully. It seems that focusing on (d) + + ′ ′ non-commutativity of operators Yi may lead to better γbasic = Φ Φ AB ρA B . (288) | ih | ⊗ understanding of the properties of this class of states. d 1 1 with94. Φ+ = − ii . Namely the following con- | i i=0 √d | i trolled transformation, called twisting, P ′ ′ 4. Purity and correlations: how they are present in p-bit U = kl kl U A B (289) τ | iABh |AB ⊗ kl Xkl In the introductory paragraph to Sec. XIX we have 95 pointed out that the origin of private correlations in transforms γbasic into a private state γ given by (287) . quantum mechanics is the existence of pure state which In general the unitary Uτ can be a highly nonlocal oper- ation, in the sense that it cannot be inverted by means of still exhibits correlations (unlike in classical world). How- local operations and classical communication. Thus the ever, we have argued then that there are states which ex- Φ+ is contained somehow in the private state, but in a hibit private key, but are quite mixed. They can have ar- | i + bitrarily small distillable entanglement, still offering bits “twisted” way: Therefore, unlike in the Φ state, ED | i of perfect key. One might wonder if it implies that we can be much smaller than EC , as shown by the following example (Horodecki et al., 2005e): need a more general principle, which explains privacy ob- tained in quantum mechanics. There is no simple solu- + + ργ = p φ φ ρs + (1 p) φ− φ− ρa (290) tion of this problem. Let us first argue, that although in | ih |⊗ − | ih |⊗ different context, the principle of ”purity & correlations” where φ± are the maximally entangled states, ρs,a are still works here. normalized| i projectors onto symmetric and antisymmetric To this end, let us note that private state is a uni- subspaces and p = 1 (1 1 ). One computes logarithmic 2 − d tarly rotated basic p-bit. The latter is of the form negativity, which bounds ED, to obtain γ = Φ+ Φ+ ρ ′ ′ (see Eqs. (288) and (289)). basic | ih |AB ⊗ A B 1 The unitary Uτ might melt completely the division of ED(ργ ) log(1 + ). (291) ≤ d γbasic into subsystems A, B, A′ and B′. However what is for sure preserved, is that the private state is a product of some pure state ψ and ρ across some tensor product. In other words, there is a virtual two-qubit subsystem, 92 More precisely, the class of private states is locally equivalent to which is in singlet state ψ = φ+ . (This virtual singlet (d) virt the one given by equation (287), that is contains any state γ is what we call “twisted” version of the original singlet rotated by local unitary transformation UA UB IA′B′ . 93 ⊗ ⊗ state.) Because of purity, the outcomes of any measure- Dimensions dA′ and dB′ are in principle arbitrary (Without los- ing generality, we can put them equal). ment performed on this virtual subsystem are not known 94 In this section we will refer to the state Φ+ as well as φ+ = to Eve. Thus we already have one ingredient of the prin- 1 ( 00 + 11 ) as to singlet state. | i | i √2 | i | i ciple — the role of purity. Let us now examine the second 95 The unitaries Ui and Uj† in definition of private state corresponds ingredient — correlations. The pure state is no longer a to Ukk and Ull† in definition of twisting. state of two qubits, one of them on Alice side, and the 87 second on Bob’s. The measurements that access individ- bound involving the best separable approximation ually the virtual qubits are the original local measure- (Moroder et al., 2006b) which exploits the fact that ments of Alice and Bob, rotated however by the inverse admixing separable state can only decrease KD. In of unitary Uτ . Now, it happens that local measurements (Moroder et al., 2006a) there is also a bound for one- on key part (AB subsystem) of the private state in stan- way distillable key, based on the fact that for a state dard basis commute with Uτ (289), for the latter is a which has a symmetric extension, its one-way distillable unitary controlled just by this basis. Thus at least one key must vanish. Indeed, then Bob and Eve share with correlation measurement on the virtual qubits is still ac- Alice the same state, so that any final key which Alice cessible to Alice and Bob, and can be performed on the shares with Bob, she also share with Eve. qubits A and B. It is nevertheless natural quite natural to maintain that the ”purity & correlations” is too strong principle for en- 6. Drawing secure key from bound entanglement. suring security. Simply, the purity of a virtual subsys- tem is not the purity of the total state that Alice and Bound entanglement is a weak resource, especially in Bob have. In the case of basic p-bit, it is easy to obtain the bipartite case. For a long time the only useful task the whole system in a pure state, simple by removing the that bipartite BE states were known to perform was ac- mixed subsystem A′B′. However, for a generic private tivation, where they acted together with some distillable state, the purity is inaccessible: the state of the whole state. Obtaining private key from bound entanglement, system is irrevocably mixed. This is because Alice and a process which we will present now, is the first useful Bob have only local access to the system, hence from task which bipartite BE states can do themselves. some p-bits Alice and Bob may be able to distill only an Since distillable entanglement of some private states amount of purity much smaller than the amount of pri- can be low it was tempting to admix with small prob- vacy they can get. Even more: if we allow arbitrary small ability some noise in order to obtain a state which is inaccuracy, then they may not be able to obtain a sys- non-distillable while being still entangled: tem in a pure correlated state at all, while still obtaining secure key. ρtotal = (1 p)γ + pρnoise (296) Thus to get privacy, one does not need to share pure − states. There is a simple explanation how it can be so: It happens that for certain private states γ the state A pure state Φ+ offers secure key in any correlated basis, ρnoise can be adjusted in such a way that the state ρtotal while what we need is just a single secure basis, because is PPT (hence E = 0), and despite this, from many we need only classical secure key - pair of bits. D copies of ρtotal one can distill key of arbitrarily good qual- ity. That is one can distill private state γ′ with arbitrary small admixture of noise. 5. Distillable key as an operational entanglement measure The first examples of states with positive distillable key and zero distillable entanglement were found in The concept of private states allows to represent KD as a quantity analogous to entanglement distillation: (Horodecki et al., 2005b,e). We present here a simple one which has been recently given in (Horodecki et al., 2005f). It is actually a mixture of two private bits (cor- LOCC key related and anticorrelated). The total state has a matrix n distillation/ (d) form ρAB⊗ γABA′B′ , (294) p1 X1 0 0 p1X1 where the highest achievable ratio log d in the asymptotic | | n 1 0 p2 X2 p2X2 0 ′ ′ limit equals the distillable key denoted as KD(ρAB). In- ρABA B = | | ,(297) 2 0 p2X† p2 X2 0 stead of singlets we distill private states. Since the class 2 | | p X† 0 0 p X of private states is broader than the class of maximally 1 1 1| 1| entangled states, one can expect that distillable key can 1 ′ ′ be greater than distillable entanglement. Indeed, this is with X1 = i,j=0 uij ij ji A B and X2 = 1 | ih | the case, and an example is just the private state of Eq. u ii jj ′ ′ where u are the elements of 1 i,j=0 ij| ih |APB ij (290). qubit Hadamard transform and p = √2 (p =1 p ). Thus distillable key is a new operational measure of en- P 1 1+√2 2 − 1 This state is invariant under partial transposition over tanglement, which is distinct from ED. It is also distinct from E and it satisfies Bob’s subsystem. If we however project its key part C (AB subsystem) onto a computational basis it turns ED KD EC . (295) out that the joint state of Alice, Bob and Eve system ≤ ≤ is fully classical and of very simple form: with proba- Moreover it is upper bounded by relative entropy of bility p1 Eve knows that Alice and Bob are correlated, entanglement (Horodecki et al., 2005e) and squashed while with probability p2 that they are anticorrelated. entanglement (Christandl, 2006). There is also a Thus the mutual information I(A : E) = 0, and 88 I(A : B)=1 H(p1). Thus applying Devetak-Winter applications, since the eavesdropper can interrupt the protocol96 (see− formula (284)) we obtain a key rate communication and entangle copies of the state. It was then unclear whether one can obtain gap between dis- KD(ρbe) 1 h(p1)=0.0213399 > ED(ρbe)=0. (298) tillable key and distillable entanglement (as reported ≥ − in Sec. XIX.B.5) in the general scenario, where Al- Basing on this example it is argued (Horodecki et al., ice and Bob do not know a priori anything about their 2005f), that the volume of bound entangled key distill- states. It hasn’t been noticed, that a positive answer able states is non-zero in the set of states occupying more to this question follows from the results on finite Quan- then 4 qubits. It is however a nontrivial task to provide tum de Finetti theorem by R. Renner, N. Gisin and B. new examples. Interestingly, no previously known bound Kraus (Kraus et al., 2005; Renner et al., 2005) (see es- entangled state has been shown to be key-distillable. pecially (Renner, 2005) and (Koenig and Renner, 2005)) and the results of (Horodecki et al., 2005e) on bound entangled key distillable states. In the meantime, a C. Private states — new insight into entanglement theory more entanglement-based approach has been developed of mixed states (Horodecki et al., 2006a,c), which can be seen as a gener- alization of Lo and Chau entanglement purification based Investigations concerning distillable key were fruitful approach to the private state distillation one. It has to entanglement theory itself. A new operational mea- been shown there, that an unconditionally secure key can sure of entanglement was obtained, and also a new source be arbitrarily greater than the amount of entanglement of examples of irreversibility in entanglement distillation which can be distilled, and even when the latter is zero. was provided. The private states, or PPT states with This important result can be rephrased as follows: nonzero key, constitute a new zoo of states which are There are situations in which one can not send easy to deal with and have nontrivial properties, in addi- • tion to such canonical classes as Werner, isotropic, Bell faithfully any qubit, but one can send arbitrarily diagonal states or maximally correlated states. While many unconditionally secure bits. the simplicity of the latter classes comes from symme- tries (e.g. invariance under twirling), simplicity of the class of private states is based just on special asymmetry 1. “Twisting” the standard protocol between the key and the shield part. The private bits called flower states ( given by trace The situation is as follows. Alice and Bob have quan- over subsystem E of (272)) are the ones for which the tum channel Λ, which can only produce bound entangled squashed entanglement has been computed. Interest- states (this is so called binding entanglement channel). ingly, in this case E = E . Moreover basing on this Clearly, quantum information cannot be sent faithfully sq C via such channel: otherwise it would be possible to trans- family one can find states with EC arbitrarily greater than the squashed entanglement and the latter arbitrar- mit faithfully singlets(see Sec. XIV). Suppose further that if Alice sends half of singlet, then, provided Eve ily greater than ED (Christandl and Winter, 2003). The flower states exhibit locking (see Sec. XVIII.A). There has not changed the channel, they obtain a state ρ of is actually a general link between locking effect and the type (297). (i.e. the state ρ and the channel Λ are con- problem of drawing key from bound entanglement. Last nected with each other via Choi-Jamio lkowski isomor- phism (I Λ) Φ+ Φ+ = ρ). The state has the property but not least, the description of this class of states yields a ⊗ | ih | natural generalization of pure maximally entangled states that after measuring the key part in standard basis and to the case of mixed states with coefficients becoming op- processing the outcomes classically, Alice and Bob obtain erators. secure key. Since Eve might change the channel, standard ap- proach would be to check bit error rate and phase error rate. Unfortunately, this would not work: the channel D. Quantum key distribution schemes and security proofs itself (without any action of Eve) produces too high er- based on distillation of private states - private key beyond purity rors. Indeed, low error rates would mean that the state is close to singlet, which is not the case: the state ρ is Key distillation described in Sec. XIX.B relies upon bound entangled hence its fidelity with singlet is no bet- important assumption. The initial state shared by Al- ter than that for separable (i.e. key undistillable) state ice and Bob should be tensor product of the same state (see Sec. VI.B.3). To overcome this problem, we have to use the fact ρAB. This assumption is unreal in almost all security that (as discussed in Secs. XIX.B.2 and XIX.B.3) private states contain perfect singlet twisted by some global uni- tary transformation. Therefore the present state, which 96 Since in this particular case the state is fully classical, it would is a noisy version of private state, contains a twisted noisy be enough to use the classical predecessor of Devetak-Winter singlet. Should Alice and Bob have access to the twisted protocol, called Csiszar-K¨orner-Maurer protocol. noisy singlet, they would estimate its quality by measur- 89 ing on selected number of systems the observables σ σ quantum secret sharing, third man quantum cryptogra- z ⊗ z (bit error) and σx σx (phase error). In the case of phy), but because it disallows certain schemes (like quan- twisted singlets, they⊗ can still estimate directly bit error tum bit commitment, quantum one-out-of-two oblivious by measuring the observable σ σ on the key part of the transfer). z ⊗ z shared states. However the observable σx σx does not commute with twisting, so that to estimate⊗ phase error of a twisted noisy singlet means to estimate a nonlocal 1. Impossibility of quantum bit commitment — when observable. Fortunately, this can be done, by decompos- entanglement says no ing the observable into local ones, and measuring them separately 97. Historically, it was claimed that QIT can ensure not The rest of the protocol, is that Alice and Bob mea- only unconditionally secure key distribution but also a sure the key part of the remaining systems (i.e. not very important ingredient of classical cryptographic pro- used for error testing), and performs standard error cor- tocols — a bit commitment protocol (Brassard et al., rection/privacy amplification procedures based on esti- 1993). If so, Alice could commit some decision (a bit mated error level. How to see that such protocol is se- value) to Bob, so that after committing she could not cure? One way is to notice, that it does the same to change her mind (change the bit value) but Bob also twisted state, as the standard BB84-like protocol (such could not infer her decision before she lets to open it. as e.g. (Ardehali et al., 1998; Lo et al., 2005)) does to Such a protocol would be one of the most important the usual state. How this looks like in terms of entan- ingredients in secure transaction protocols. Unfortu- glement purification? The standard protocol is secure, nately, it is not the case: Mayers (Mayers, 1996, 1997) because it would produce singlets. The present protocol and independently Lo and Chau (Lo and Chau, 1997, is secure, because, if performed coherently it produces 1998) have proved under assumptions plausible in cryp- private state, i.e. it is twisted purification. Of course, tographic context, that quantum bit commitment is not one can easily convert it into P&M protocol along Shor- possible. Paradoxically, it is exactly entanglement, which Preskill lines. assures security of QKD , that is the main reason for which the quantum bit commitment is not possible. It We have arrived at general principle, which is actually shows the following important fact: When the two par- “if and only if”: A protocol produces secure key if and ties do not trust each other, entanglement between them only if its coherent version produces private states. may sometimes become the most unwanted property. This statement connects entanglement and key dis- There were many attempts to perform quantum bit tribution in an ultimate way. It was recently applied commitment; some of them invalid as covered by the by Renes and Smith (Renes and Smith, 2007) who have proof given by Lo and Chau and some of them being found an entanglement based proof of the P&M pro- approximated versions of impossible quantum bit com- tocol with noisy preprocessing of (Kraus et al., 2005; mitment. Renner et al., 2005). They have demonstrated its coher- While the proof of Lo and Chau is valid, as it was ent version which distills private states, and hence must pointed out by H. P. Yuen (Yuen, 2005) one could weaken be secure. assumptions, so that the Lo-Chau theorem does not ap- We have quite a paradoxical situation. When it turned ply. Namely, the initial state of Bob in this two-party out that one can draw secure key in the situation where protocol may not be fixed at the beginning. It leads to no singlets can be distilled, it seemed natural that to considerations similar to a game theoretic situation. This prove unconditional security, one cannot use the tech- case is almost covered by the latest result of D’Ariano et niques based on entanglement purification. Surprisingly, al. (D’Ariano et al., 2006). The latter paper also pro- it is not the case: everything goes through, in a twisted vides the most recent and wide review of this topic. form. It is important to note, that the same impossibil- ity reasons share other desired protocols such as ideal quantum coin tossing, one-out-of-two oblivious transfer E. Entanglement in other cryptographic scenarios and one-sided two party secure computations (Lo, 1997; Lo and Chau, 1998). There are many other quantum cryptographic scenar- ios than quantum key agreement, where entanglement enters. Here we comment briefly on some of them. Inter- 2. Multipartite entanglement in quantum secret sharing estingly, this time entanglement is important not only because it makes some protocols possible (like QKD, There are situations in which one does not trust a sin- gle man. A secret should be shared by a few people so that no single person could know the truth without per- mission of the others. Classically, splitting information 97 One needs quantum de Finetti theorem to prove that such sep- into pieces among parties, so that each piece is informa- arate estimation will report correctly the value of global observ- tionless unless they cooperate, is known as secret sharing able. (Blakley, 1979; Shamir, 1979). 90 Quantum secret sharing protocol, as proposed in using this distributes a single key, which all the par- (Hillary et al., 1999), solves the following related prob- ties will share finally. This however can be achieved lem, which can be viewed as secure secrete sharing. Char- in a much simpler way using a multipartite entangled lie wants to perform secret sharing, but is far from Alice state. A state which is mostly used for this task is the and Bob, thus he has to face the additional problem - n partite GHZ state (Chen and Lo, 2004). One can also eavesdropping. He could resolve it if he had two separate use for this task multipartite version of private states secret channels with them, sending to one person a key, (Horodecki and Augusiak, 2006). and to the other the encoded message (secret). However Another interesting application of multipartite entan- there is a more direct way discovered in (Hillary et al., gled states is the so called third man quantum cryptog- 1999), using GHZ state raphy (Zukowski˙ et al., 1998b). This cryptographic sce- | i nario involves traditionally Alice and Bob, and a third 1 person Charlie, who controls them. Only when Charlie GHZ ABC = ( 000 + 111 ) (299) | i √2 | i | i wants, and tells them some information, they can share a quantum private channel and they verify, that indeed no- In their protocol all three parties are supplied many body including Charlie have access to this channel. This copies of the above multipartite state. Then each of can be easily achieved by employing the GHZ state them measures randomly either in + , basis, or in and the idea of “entanglement with assistance”| (seei Sec. 1 1 {| i |−i} ( 0 +i 1 ), ( 0 i 1 ) . After this step each party XV). Any two-qubit subsystem of GHZ state is in sep- { √2 | i | i √2 | i− | i } announces the basis that was chosen, but not the result. arable state, hence useless for cryptography.| i However if In half of the cases, Alice and Bob can infer the result of Charlie measures a third qubit in + , basis, and Charlie’s measurement when they meet and compare the tells Alice and Bob holding other two{| qubitsi |−i} the result of results. Thus the results of Charlie’s measurement, which his measurement then they obtain one of the maximally are random, can serve as a key, and Alice and Bob can entangled states. Then Alice and Bob can verify that receive its split copy. Moreover it can be shown, that this they indeed share the entangled states and use them to protocol is secure against any eavesdropping and even a launch a QKD protocol. cheating strategy of either Alice or Bob. We have discussed sharing of classical information by means of quantum entanglement. One can also share F. Interrelations between entanglement and classical key quantum information, and entanglement again helps agreement (Hillary et al., 1999). This time Charlie wants to split securely a qubit, so that Alice and Bob would need to So far we have discussed the role of entanglement in cooperate to recover it. Interestingly, sharing a quan- quantum cryptography. It is interesting, that entangle- tum secret, is essentially teleportation of a qubit through ment, which is originally quantum concept, corresponds a GHZ state. After Charlie performs teleportation, the to privacy in general - not only in the context of quan- qubit is split between Alice and Bob. To reconstruct the tum protocols. Here the interaction between entangle- qubit one of the parties (i.e. Alice) measures his qubit in ment theory and the domain of classical cryptography + , basis and sends the result of his measurement called classical key agreement (CKA) is presented. to{| thei |−i} other party (i.e. Bob). Finally the other party applies on of two unitary operations. The problem of distilling secret key from correlations shared by Alice and Bob with presence of an eaves- This scheme has been further developed within dropper Eve was first studied by Wyner (Wyner, 1975) the framework of quantum error correcting codes in and Csiszar and Korner (Csiszar and K¨orner, 1978). It (Cleve et al., 1999; Gottesman, 2000). Interestingly, the was introduced as a classical key agreement scenario and protocol of Hillery at al. can be changed so that it uses studied in full generality by Maurer (Maurer, 1993). Ac- bipartite entanglement only (Karlsson et al., 1999). This cording to this scenario, Alice and Bob have access to simplification made it possible to implement the scheme n independent realizations of variables A and B respec- in 2001 (Tittel et al., 2001). For the recent experimen- tively, while the malicious E holds n independent realiza- tal realization see (Chen et al., 2005c), and references tions of a variable Z. The variables under consideration therein. have joint probability distribution P (A, B, E). The task of Alice and Bob is to obtain via local (classical) op- erations and public communication (LOPC) the longest 3. Other multipartite scenarios bit-string which is almost perfectly correlated and about which Eve (who can listen to the public discussion) knows One of the obvious generalization of quantum key dis- a negligible amount of information98. tribution is the so called conference key agreement. When some n parties who trust each other want to talk securely, so that each of them could receive the information, they could do the following: one party makes n 1 QKD “bi- 98 − Let us emphasize, that unlike in quantum cryptography, it is in partite” protocols with all the other parties, and then principle not possible to bound Eve’s information on classical 91 Here the probability distribution P (A, B, E) is a priori entanglement theory key agreement given. I.e. it is assumed, that Alice and Bob somehow quantum secret classical know how Eve is correlated with their data. entanglement correlations quantum secret classical 1. Classical key agreement — analogy to distillable communication communication entanglement scenario classical public classical Classical key agreement scenario is an elder sibling of communication communication the entanglement distillation-like scenario. This relation was first found by N. Gisin and S. Wolf (Gisin and Wolf, local actions local actions 1999, 2000), and subsequently developed in more general- ity by Collins and Popescu (Collins and Popescu, 2002). TABLE I Here we present relations between basic no- tions of key agreement and entanglement theory following The analogy has been explored in the last years and (Collins and Popescu, 2002) proved to be fruitful for establishing new phenomena in classical cryptography, and new links between pri- vacy and entanglement theory. The connections are quite 1996c,d) have been designed on basis of protocols of clas- beautiful, however they still remain not fully understood sical key agreement. The feedback from entanglement by now. theory to classical key agreement was initiated by Gisin The classical key agreement task is described by the and Wolf (Gisin and Wolf, 2000) who asked the question, following diagram: whether there is an analogue of bound entanglement, which we discuss in next section. Subsequently, in anal- ogy to entanglement cost which measures how expensive classical distillation of in terms of singlet state is the creation of a given quan- key tum state ρAB by means of LOCC operations Renner and n / k [P (A, B, E)]⊗ [P (K,K,E′)]⊗ , (300) Wolf (Renner and Wolf, 2003) have defined information of formation denoted as I (A; B E) (sometimes called form | where P (K,K,E′) is perfectly secure distribution satis- “key cost”). This function quantifies how many secure fying: key bits (301) the parties have to share so that they could create given distribution P (A, B, E) by means of LOPC 1 P (K,K) P (i, j)= δij operations. The axiomatic approach to privacy, resulting ≡{ 2 } in deriving secrecy monotones (also in multipartite case), P (K,K,E′)= P (K,K)P (E′) (301) has been studied in (Cerf et al., 2002b; Horodecki et al., 2005c). where Alice and Bob hold variable K and E′ is some An interesting formal connection between CKA and Eve’s variable, i.e. Alice and Bob are perfectly correlated entanglement is the following (Gisin and Wolf, 2000). and product with Eve. The optimal ratio k in asymptotic n Any classical distribution can be obtained via POVM limit is a (classical) distillable key rate denoted here as measurements on Alice, Bob and Eve’s subsystems of a K(A; B E) (Maurer, 1993). pure quantum state ψ : Entanglementk between two parties (see Sec. XIX) re- | iABE ports that nobody else is correlated with the parties. In ABE (i) (j) (k) p := TrM M M ψ ψ ABE (302) similar way the privacy of the distribution P (A, B, E) ijk A ⊗ B ⊗ E | ih | means that nobody knows about (i.e. is classically cor- (.) where MA,B,E are elements of POVM’s of the parties sat- related with) the variables A and B. In other words, any (l) tripartite joint distribution with marginal P (A, B) has a isfying l MA,B,E =IA,B,E with I the identity operator. product form P (A, B)P (E). Conversely, with a given tripartite distribution P Following along these lines one can see the nice cor- P (A, B, E) one can associate a quantum state in the fol- respondence between maximally entangled state φ+ = lowing way: 1 | i √ ( 00 + 11 ) and the private distribution (301), and 2 | i | i P (ABE)= pABE ψ = pABE ijk . also the correspondence between the problem of transfor- { ijk } 7−→ | ABEi ijk | iABE n ijk mation of the state ρAB⊗ into maximally entangled states X q which is the entanglement distillation task and the above (303) described task of classical key agreement. Actually, the According to this approach, criteria analogous to first entanglement distillation schemes (Bennett et al., those for pure bipartite states transitions and cat- alytical transitions known as majorization criteria (see Secs. XIII.A and XIII.A.1) can be found (Collins and Popescu, 2002; Gisin and Wolf, 2000). Also ground. However, in particular situations, there may be good other quantum communication phenomena such as merg- practical reasons for assuming this. ing (Horodecki et al., 2005h) and information exchange 92 (Oppenheim and Winter, 2003) as well as the no-cloning and Bob’s variables, such that the resulting distribution principle are found to have counterparts in CKA has nonzero key (see in this context (Acin et al., 2003a,b; (Oppenheim et al., 2002b). Bruß et al., 2003)). A simple and important connection between tripartite A strong confirmation supporting hypothesis of bound distributions containing privacy and entangled quantum information is the result presented in (Renner and Wolf, states was established in (Acin and Gisin, 2005). Con- 2003), where examples of distributions which asymp- sider a quantum state ρ and its purification ψ totically have bound information were found. Namely AB | ABEi to the subsystem E. there is a family of distributions P (An,Bn, En) such that 1 limn K(An; Bn En) = 0 while Iform(An; Bn En) > If a bipartite state ρ is entangled then there ex- →∞ k | 2 • AB for all n. Another argument in favor of the existence of ists a measurement on subsystems A and B such, bound information in this bipartite scenario, is the fact that for all measurements on subsystem E of purifi- that the multipartite bound information has been already cation ψ the resulting probability distribution | ABEi proved to exist, and explicit examples have been con- P (A, B, E) has nonzero key cost. structed (Acin et al., 2004b). More specifically this time three honest cooperating parties (Alice, Bob and Clare) If a bipartite state ρ is separable, then for all AB and the eavesdropper (Eve) share n realizations of a joint • measurements on subsystems A and B there ex- distribution P (A,B,C,E) with the following prop- ists a measurement subsystem E of purification bound erties: it has nonzero secret key cost and no pair of honest ψ such that the resulting probability distri- ABE parties (even with help of the third one) can distill secret bution| iP (A, B, E) has zero key cost. key from these realizations P (A,B,C,E)bound. Its construction has been done according to a Gisin- Wolf procedure (302) via measurement of a (multipartite) 2. Is there a bound information? purification of a bound entangled state in computational basis. In the entanglement distillation scenario there are As an example consider a four partite state bound entangled states which exhibit the highest irre- versibility in creation-distillation process, as the distill- 1 1 able entanglement is zero although the entanglement cost ψbe = [ GHZ + ( 001 1 | i √ | i √ | i| i is a nonzero quantity(see Sec. XII). One can ask then if 3 6 + 010 2 + 101 3 + 110 4 )] (304) the analogous phenomenon holds in classical key agree- | i| i | i| i | i| i ABCE ment called bound information (Gisin and Wolf, 2000; where GHZ = 1 ( 000 + 111 ). It’s partial trace over Renner and Wolf, 2003). This question can be stated as | i √2 | i | i follows: subsystem E is a multipartite bound entangled state99 (D¨ur and Cirac, 2000b; D¨ur et al., 1999b). If one mea- Does there exist a distribution P (A, B, E)bound for sures ψbe in computational basis on all four subsystems, • which a secure key is needed to create it by LOPC the resulting| i probability distribution on the labels of out- (Iform(A; B E) > 0), but one cannot distill any key comes is the one which contains (tripartite) bound infor- back from it| (K(A; B E)=0)? mation (Acin et al., 2004b). k In (Gisin and Wolf, 2000) the tripartite distributions obtained via measurement from bound entangled states XX. ENTANGLEMENT AND QUANTUM COMPUTING were considered as a possible way of search for the hy- pothetical ones with bound information. To get Eve’s A. Entanglement in quantum algorithms variable, one has first to purify a bound entangled state, and then find a clever measurement to get tripartite dis- Fast quantum computation is one of the most desired tribution. In this way, there were obtained tripartite dis- properties of quantum information theory. There are tributions with non-zero key cost. However the no-key few quantum algorithms which outperform their classical distillability still needs to be proved. counterparts. These are the celebrated Deutsch-Jozsa, Yet there are serious reasons supporting conjec- Grover and Shor’s algorithm, and their variations. Since ture that such distributions exist (Gisin et al., 2002; entanglement is one of the cornerstones of quantum in- Renner and Wolf, 2002). To give an example, in formation theory it is natural to expect, that it should (Renner and Wolf, 2002) an analogue of the necessary be the main ingredient of quantum algorithms which are and sufficient condition for entanglement distillation was better than classical. This was first pointed out by Jozsa found. As in the quantum case the state is distillable iff in (Jozsa, 1997). His seminal paper opened a debate on there exists a projection (acting on n copies of a state for some n) onto 2-qubit subspace which is entangled (Horodecki et al., 1998a), in the classical case, the key is distillable iff there exists a binary channel (acting on n 99 This is one of the states from the family given in Eq. (86), with + 1 1 copies of a distribution for some n) which outputs Alice’s parameters: m = 3, λ0 = 3 , λ0− = λ10 = 0, λ01 = λ11 = 6 . 93 the role of entanglement in quantum computing. Actu- useful in quantum computing. Answering to the general ally, after more than decade from the discovery of the question of how big the enhancement based on separa- first quantum algorithm, there is no common agreement ble states may be, needs more algorithm-dependent ap- on the role of entanglement in quantum computation. proach. We discuss major contributions to this debate. It seems That the presence of entanglement is only neces- that entanglement “assists” quantum speed up, but is sary but not sufficient for exponential quantum speed not sufficient for this phenomenon. up follows from the famous Knill-Gottesman theo- Certainly pure quantum computation needs some level rem (Gottesman and Chuang, 1999; Jozsa and Linden, of entanglement if it is not to be simulated classically. 2002). It states that operations from the so called Clif- It was shown by Jozsa and Linden, that if a quantum ford group composed with Pauli measurement in com- computer’s state contains only constant (independent of putational basis can be efficiently simulated on classical number of input qubits n) amount of entanglement, then computer. This class of operations can however produce it can be simulated efficiently (Jozsa and Linden, 2002). highly entangled states. For this reason, and as indicated Next, Vidal showed that even a polynomial in n by other results cited above, the role of entanglement is amount of entanglement present in quantum algorithm still not clear. As it is pointed out in (Jozsa and Linden, can also be simulated classically (Vidal, 2003). The re- 2002), it may be that what is essential for quantum com- sult is phrased in terms of the number χ which he defined putation is not entanglement but the fact that the set of as the maximal rank of the subsystem of the qubits that states which can occur during computation can not be form quantum register of the computer (over all choices described with small a number of parameters (see also of the subsystem). Any quantum algorithm that main- discussion in (Knill, 2001) and references therein). tains χ of order O(poly(n)) can be efficiently classically simulated. In other words to give an exponential speedup the quantum algorithm needs to achieve χ of exponential B. Entanglement in quantum architecture order in n, during computation. This general result was studied by Orus and La- Although the role of entanglement in algorithms is torre (Orus and Latorre, 2004) for different algorithms in unclear, its role in architecture of quantum computers terms of entropy of entanglement (von Neumann entropy is crucial. First of all the multipartite cluster states of subsystem). It is shown among others that computa- provide a resource for one-way quantum computation tion of Shor’s algorithm generates highly entangled states (Raussendorf and Briegel, 2001). One prepares such (with linear amount of entropy of entanglement which multipartite state, and the computation bases on subse- corresponds to exponential χ). Although it is not known quent measurements of qubits, which uses up the state. if the Shor’s algorithm provides an exponential speedup One can ask what other states can be used to per- over classical factoring, this analysis suggests that Shor’s form universal one-way quantum computation. In algorithm cannot be simulated classically. (den Nest et al., 2006) it was assumed that universality Entanglement in Shor’s algorithm has been stud- means possibility of creating any final state on the part ied in different contexts (Ekert and Jozsa, 1998; of lattice that was not subjected to measurements. It Jozsa and Linden, 2002; Parker and Plenio, 2002; was pointed out that by use of entanglement measures Shimoni et al., 2005). Interestingly, as presence of one can rule out some states. Namely, they introduced entanglement in quantum algorithm is widely con- an entanglement measure, entanglement width, which is firmed (see also (Datta et al., 2005; Datta and Vidal, defined as the minimization of bipartite entanglement 2006)), its role is still not clear, since it seems that entropy over some specific cuts. It turns out that this amount of it depends on the type of the input numbers measure is unbounded for cluster states (if we increase (Kendon and Munro, 2006b). size of the system). Thus any class of states, for which Note, that the above Jozsa-Linden-Vidal “no entan- this measure is bounded, cannot be resource for universal glement implies no quantum advantage on pure states” computation, as it cannot create arbitrary large cluster result shows the need of entanglement presence for ex- states. For example the GHZ state is not universal, since ponential speed up. Without falling into contradiction, under any cut the entropy of entanglement is just 1. One one can then ask if entanglement must be present for should note here, that a more natural is weaker notion polynomial speed up when only pure states are involved of universality where one requires possibility of compute during computation (see (Kenigsberg et al., 2006) and arbitrary classical function. In (Gross and Eisert, 2006) references therein). it is shown that entanglement width is not a necessary Moreover it was considered to be possible, that a condition for this type of universality. quantum computer using only mixed, separable states An intermediate quantum computing model, be- during computation may still outperform classical ones tween circuit based on quantum gates, and one- (Jozsa and Linden, 2002). It is shown, that such phe- way computing, is teleportation based computing nomenon can hold (Biham et al., 2004), however with a (Gottesman and Chuang, 1999). There two and three tiny speed up. It is argued that isotropic separable state qubit gates are performed by use of teleportation as a cannot be entangled by an algorithm, yet it can prove basic primitive. The resource for this model of computa- 94 tion are thus EPR states and GHZ states. Teleportation C. Byzantine agreement — useful entanglement for based computing is of great importance, as it allows for quantum and classical distributed computation efficient computation by use of linear optics (Knill et al., 2001), where it is impossible to perform two-qubit gates As we have already learned the role of entanglement in deterministically. Moreover, using it, Knill has signifi- communication networks is uncompromised. We have al- cantly lowered the threshold for fault tolerant computa- ready described its role in cryptography (see Sec. XIX) tion (Knill, 2004). and communication complexity. Here we comment an- other application - quantum solution to one of the famous The need of entanglement in quantum architecture has problem in classical fault-tolerant distributed computing been also studied from more “dynamical” point of view, called Byzantine agreement. This problem is known to where instead of entangled states, one asks about opera- have no solution in classical computer science. Yet its tions which generate entanglement. This question is im- slightly modified version can be solved using quantum portant, as due to decoherence, the noise usually limits entangled multipartite state (Fitzi et al., 2001). entanglement generated during the computation. This One of the goals in distributed computing is to achieve effect can make a quantum device efficiently simulatable broadcast in situation when one of the stations can send and hence useless. To avoid this, one has to assure that faulty signals. The station achieve broadcast if they fulfill operations which constitute a quantum device generate conditions which can be viewed as a natural extension of sufficient amount of (or a robust kind of) entanglement. broadcast to a fault-tolerant scenario: 1. if the sending station is not faulty, all stations an- An interesting connection between entanglement and nounces the sent value. fault tolerant quantum computation was obtained by Aharonov (Aharonov, 1999). She has shown that a prop- 2. if the sending station is faulty, then all stations erly defined long range entanglement 100 vanishes in ther- announce the same value. modynamical limit if the noise is too large. Combining it with the fact that fault tolerant scheme allows to achieve It is proved classically, that if there are t n/3 sta- such entanglement, one obtains a sort of phase transition tions which are out of work and can send unpredictable≥ (the result is obtained within phenomenological model of data, then broadcast can not be achieved. In quantum noise). scenario one can achieve the following modification called detectable broadcast which can be stated as follows: The level of noise under which quantum computer becomes efficiently simulatable was first studied in 1. if there is no faulty station, then all stations an- (Aharonov and Ben-Or, 1996). It is shown that quantum nounces the received value. computer which operates (globally) on O(log n) number 2. if there is faulty station, then all other stations ei- of qubits at a time (i.e. with limited entangling capabili- ther abort or announces the same value. ties) can be efficiently simulated for any nonzero level of noise. For the circuit model (with local gates), the phase This problem was solved in (Fitzi et al., 2001) for n = transition depending on the noise level is observed (see 3. We do not describe here the protocol, but comment on also (Aharonov et al., 1996)). the role of entanglement in this scheme. In the quantum Byzantine agreement there are two stages: quantum and The same problem was studied further in classical one. The goal of quantum part is to distribute (Harrow and Nielsen, 2003), however basing directly among the stations some correlations which they cannot on the entangling capabilities of the gates used for deny (even the one which is faulty). To this end, the computation. It is shown, that so called separable gates parties perform a protocol which distributes among them (the gates which can not entangle any product input) the Aharonov state: are classically simulatable. The bound on minimal noise 1 level which turns a quantum computer to deal only ψ = ( 012 + 201 + 120 021 102 210 ) (305) with separable gates is provided there. This idea has √6 | i | i | i−| i−| i−| i been recently developed in (Virmani et al., 2005), where certain gates which create only bipartite entanglement The trick is that it can be achieved fault-tolerantly, i.e. are studied. This class of gates is shown to be classically even when one of the stations sends fake signals. simulatable. In consequence, a stronger bound on the tolerable noise level is found. ACKNOWLEDGMENTS We would like to thank J. Horodecka for her continu- 100 Another type of long range entanglement was recently defined ous help in editing of this paper. Many thanks are due (Kitaev and Preskill, 2006; Levin and Wen, 2006) in the context to A. Grudka,L. Pankowski, M. Piani and M. Zukowski˙ of topological order. for their help in editing and useful discussions. We would 95 like also to express thanks to many other colleagues in Aharonov, D., M. Ben-Or, R. Impagliazzo, and N. Nisan, the field of quantum information for discussions and use- 1996, Limitations of noisy reversible computation, eprint ful feedback. We wish also to acknowledge the referees quant-ph/9611028. for their fruitful criticism. This work is supported by EU Akhtarshenas, S. J., 2005, J. Phys. A: Math. Gen. 38, 6777, grants SCALA FP6-2004-IST no.015714 and QUROPE. eprint quant-ph/0311166. K.H. acknowledges support of Foundation for Polish Sci- Akopian, N., N. H. Lindner, E. Poem, Y. Berlatzky, J. Avron, D. Gershoni, B. D. Gerardot, and P. M. Petroff, 2006, Phys. ence. P.H. is supported by Polish Ministry if Science and Rev. Lett 96, 130501, eprint quant-ph/0512048. Education, contract No. 1 P03B 095 29. K.H., M.H. and Alber, G., T. Beth, M. Horodecki, P. Horodecki, R.H. are supported by Polish Ministry of Science and R. Horodecki, M. R¨otteler, H. Weinfurter, R. F. 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