arXiv:quant-ph/0111064v1 11 Nov 2001 nageetwtotivkn h tt estimation. state quantum the fol- detecting invoking of the without method In entanglement direct a detection. describe entanglement we the lowing of context the in tknt etesm o h w atce) ota one that so particles), two the find dimensions for always same finite can the of be are to particle (taken each with associated r ihrt n h eopsto 1 ietyo to or For [8]. directly criteria (1) large separability mathematical sufficiently decomposition the the of one find use to either try ar and pair, [9], states product of whether sum assert convex to a as need math- written we a be view From of not. or point entangled ematical are pair each in particles with u tt siain oee,ti novsestimating involves this however, d estimation, state tum aeqatmsaedsrbdb oedniyopera- density some by described tor state quantum same ain aey sadcso rbe ihquantum with compu- problem quantum We decision of structure. form a data new entanglement. as a namely, of as tation, amount provides method it the this qubits view of entangled estimation two of an case require particular not a does and any efficient is which entanglement r fgetpatcliprac.Hr,w propose we Here, viable, importance. experimentally practical an great entanglement of quantum are of detection, the estimation for and of methods verification one experimental computers. quantum efficient is building Hence, The states for [7,8]. entangled requirements maps basic generate positive studies to of con- theory the capacity interesting the side very with mathematical revealed nections have the entanglement On of synchro- [6]. clock nization and [5] positioning standards potential quantum-enhanced frequency quan- with or better example, [4], resource for cryptography as, physical tum redis- such of new recently applications foundations commercial a been the as has of covered [1–3], development mechanics the quantum in role cant 4 ups eaegiven are we Suppose − ̺ unu nageet fe lyn signifi- a playing after entanglement, Quantum priori a hc sukon ene odcd hte the whether decide to need We unknown. is which , elprmtr of parameters real 1

α i

P and i ̺ nweg bu h unu tt.In state. quantum the about knowledge p = i

.I sasmdta h ibr spaces Hilbert the that assumed is It 1. = β k X i n

k i ≤ etiigt ieetprilsi the in particles different to pertaining emyide tr ihtequan- the with start indeed may we p d i 2 | aut fApidPyisadMteais ehia Univ Technical Mathematics, and Physics Applied of Faculty If . α i n ih ̺ oto hc r irrelevant are which of most , α ̺ direct ar fprils l nthe in all particles, of pairs i h nvriyo xod ak od xodO13U UK. 3PU, OX1 Oxford Road, Parks Oxford, of University The eekonte ecould we then known were ⊗ | ietdtcino unu entanglement quantum of detection Direct etefrQatmCmuain lrno Laboratory, Clarendon Computation, Quantum for Centre β eeto fquantum of detection i ih β i | , | 092Gda´nsk, Poland. 80-952 Pawe Horodecki l ̺ ru Ekert Artur can (1) d 1 opeeypstv hsi eas h oetnegative lowest the be can because map is into resulting This matrix the positive then density completely state any mixed turns maximally that a map depolarizing a ucett osdrol oiiemp uhta the that such Λ( Λ Tr maps of positive maximum only consider to sufficient o l oiiebtntcmltl oiiemp : Λ maps positive completely not but M positive all for fmtie fdimension of matrices of nalcpe of copies all on X iwda w osctv hscloeain:firstly, be state maps which can propriate operations: transformation measurement physical a The construct consecutive we two [7]. as maps viewed positive test on mathematical best based the as entanglement quantum ing ievleof eigenvalue ae ti ae nmteaia rpriso linear of Let properties [7]. mathematical matrices on on acting based maps is positive It to di- proposed not date. criterion and separability mathematical implementable, purely rectly albeit powerful, most indicator. rability tbcmseprmnal ibewtotivligany involving without estimation. viable state experimentally becomes it aldpstv if positive called asΛ uha nat-ntr rnpsto,adthe and transposition, anti-unitary positive an maps that as induced is snag such The Λ, choos- [10,7]. by maps transposition detected be be to example, can Λ For qubits ing two one. of just states choose entangled can we maps, positive (2). condition the affect fac- not multiplicative does positive which a tor by only differ maps positive oy hs h rtro 2 ail sue ro knowl- prior assumes of tacitly edge (2) criterion the Thus, tory. hsclyalwdtasomto fdniyoperators represents density it of (here such, transformation as allowed physically and, a positive, completely map called induced is the If trum). t rtro ed [7]: separabil- the reads terminology criterion this ity Using dimension). any of d ecntutamaueetwihcnb performed be can which measurement a construct We ovnetsatn on o u osrcini the is construction our for point starting convenient A fw i na prpit rprin[ proportion appropriate an in mix we If utemr,i oecss nta fsann all scanning of instead cases, some in Furthermore, ≥ 7→ en httematrix the that means 0 I eoe h dniympo naxlaysystem auxiliary an on map identity the denotes M ̺ oee,teei a omdf ts that so it modify to way a is there However, . d cigo h eodpril.I ati is it fact In particle. second the on acting ̺ I ′ hsegnau ln evsa sepa- a as serves alone eigenvalue This . ̺ ⊗ ′ ̺ X antb mlmne nalabora- a in implemented be cannot Λ riyo Gda´nsk of ersity n,scnl,w esr h lowest the measure we secondly, and, ̺ [ n hc sa oefli detect- in powerful as is which and vrall over ) I ≥ ⊗ ̺ mle Λ( implies 0 Λ]( ssprbeiff separable is d ealta : Λ that recall ; ̺ I ) ⊗ ≥ ̺ X sas oiiete Λ then positive also is Λ 0 seult nt.Other unity. to equal is a ongtv spec- nonnegative a has , X ) ≥ M (expression 0 ̺ M d I noa ap- an into d easpace a be ⊗ 7→ ]with Λ] M d (2) all is 2 eigenvalues generated by the induced map [(I I) (I Λ)] state ̺ is separable iff this eigenvalue satisfies λmin . ⊗ ⊗ ⊗ ≥ 9 can be offset by the positive eigenvalues of the maxi- Let us also point out an extra bonus: λmin gives us λ′, mally mixed state generated by the depolarizing map. the most negative eigenvalue of [I T ](̺), which enters− The most negative eigenvalue λ < 0 is obtained when the expression for the upper and⊗ lower bounds for the [(I I) (I Λ)] acts on the maximally− entangled state entanglement of formation, 2 ⊗ ⊗ ⊗1 d of the form 2 i=1 i i , where each state i pertains √d | i| i 1+ √1 4λ 2 to a d2 dimensional subsystem which itself is composed ′ P H − E(̺) of two d dimensional parts. Thus the map 2 ! ≤ I I 1+ 1 4 √2λ 2 + λ λ [I Λ](̺)= p ⊗ + (1 p)[I Λ](̺), (3) − ′ ′ − ′ d2 H , (9) ⊗ − ⊗ ≤  q 2
 is completelyg positive and therefore physically imple-   mentable when the induced map [(I I) (I Λ)] is where H(x) is the Shannon entropy. The above formulae positive, which happens for p (d4λ⊗)/(d4⊗λ + 1)⊗ [11]. can be derived from the estimations of the concurrence By inserting the threshold value≥ p = (d4λ)/(dg4λ + 1) provided by Verstraete et al [12]. into (3) we can modify the criterion (2) as follows: ̺ is Suppose for a moment that I Λ is trace-preserving, separable iff for all positive maps Λ, e.g. the transposition case. The⊗ first part of our en- tanglement detection measurementg is accomplished by d2λ [I Λ](̺) , (4) applying I Λ to each of the n pairs to obtain n copies ⊗ ≥ d4λ +1 ⊗ of ̺′ = [I Λ](̺). Then, following the criterion (4), we ⊗ i.e. when the minimalg eigenvalue of the transformed state need to measureg the lowest eigenvalue of ̺′. 2 4 ̺′ = [I Λ](̺) is greater than (d λ)/(d λ +1). In gen- This cang be viewed as a special case of the spectrum ⊗ eral, for some maps Λ, the related completely positive estimation and possible approaches depend a lot on par- maps IgΛ are not trace-preserving and require posts- ticular physical realizations of ̺′. Here, we provide two elections⊗ in their physical implementations. Maps such general solutions. The first one, based on quantum inter- as I Λg have been referred to as “structural” physical ferometry, is conceptually simple and relies on estimating 2 ⊗ I d 1 parameters from which the spectrum of ̺′ can be approximations of unphysical maps Λ [11]. − Forg example, if we take Λ to be transposition⊗ T , (the calculated (this is a significant gain over the state esti- mation which involves d4 1 parameters). The second first positive map used for detecting entanglement), we − obtain solution is a joint measurement on all copies of ̺′ which gives directly the estimate of the lowest eigenvalue. d 1 [I T ](̺)= I I + [I T ](̺). (5) We start with the quantum interferometry, presented ⊗ d3 +1 ⊗ d3 +1 ⊗ here as a quantum network shown in Fig.(1). A typical In theg two qubit case, where the partial transposition is interferometric set-up for a single qubit — the Hadamard a sharp test for entanglement, we obtain, gate, phase shift φ, the Hadamard gate, followed by a measurement — is modified by inserting in between the 2 1 [I T ](̺)= I I + [I T ](̺), (6) Hadamard gates a controlled-U operation, with its con- ⊗ 9 ⊗ 9 ⊗ trol on the qubit and with U acting on a quantum system which can beg represented and implemented as described by some unknown density operator ρ. (N.B. we do not assume anything about the form of ρ, it can, 1 2 for example, describe several entangled or separable sub- Λ1 Λ2 + I σxσzΛ1σzσx, (7) 3 ⊗ 3 ⊗ systems.) The action of the controlled-U on ρ modifies with the two channels defined as: the interference pattern by the factor, 1 1 iα Λ (̺)= σ ̺σ , Λ (̺)= σ ̺σ . (8) TrρU =ve , (10) 1 3 i i 2 4 i i i=x,y,z i=o,x,y,z X X where v is the new visibility and α is the shift of the The map can be implemented by applying selected interference fringes, also known as the Pancharatnam products of unitary (Pauli) transformations with the pre- phase [13]. Formula (10) has been derived, in the context scribed probabilities. The map It is trace-preserving of geometric phases, by Sj¨oqvist et al. [14]. hence any postselection in experimental realizations is avoided. Thus, in order to detect entanglement of an arbitrary two-qubit state ̺ it is enough to estimate a single pa- rameter, i.e. the minimal eigenvalue of [I T ](̺). The ⊗ g 2 2 lowest eigenvalue) of ̺′ = [I Λ](̺) by estimating d 1 k ⊗ 2 − parameters Tr ̺′ , where k =2...d . Again, the phase in the interferometry can be fixedg at ϕ = 0. The interferometric scheme described above is concep- tually simple and experimentally viable, however, if the simplicity of the implementation is not an issue then we can measure the estimate of the lowest eigenvalue di- rectly. This requires a join measurement on all of the n pairs. We use the Keyl and Werner spectrum estimation method [15], which, in the entanglement detection con- text, works as follows. The n copies of the m m state ̺′ (in our case m = d2) form an operator on the× n-fold ten- sor product space which can be decomposed according to irreps of SU(m), so that each summand, including mul- tiplicities, is labelled by a Young tableau, i.e. n boxes arranged in rows of decreasing length (c.f. [16] for the SU(2) case). The tableaus give a family of projectors for the spectrum estimation measurement. The normalized row lengths of each tableau are taken as estimates of the ordered sequence of eigenvalues of ̺′. The probability that the error is greater than some fixed ǫ decreases ex- ponentially with n [15]. In our particular case, we are FIG. 1. Both the visibility and the shift of the interfer- interested only in the lowest eigenvalue. We modify the ence patterns of a single qubit (top line) are affected by the Keyl-Werner scheme by adding together all projectors controlled-U operation. This set-up allows to estimate Tr Uρ, corresponding to Young tableaus with the fixed length the average value of U in state ρ. of the last row. The measurement determined by these projectors gives directly the estimate of only one param- The network can evaluate certain non-linear function- eter — the lowest eigenvalue of ̺′. Such a measurement als of density operators. Indeed, let us choose ρ to be can be represented as a quantum network implementing composed of two subsystems, ρ = ̺a ̺b, and let U to be projections on the symmetric and on partially symmet- ⊗ the exchange operator V such that V α β = β α ric subspaces (see [17] for the network projecting on the for any pure states of the two subsystems. The inter- symmetric subspace). ference pattern is now modified by the factor Tr V (̺ a Our considerations remain valid, with some minor ⊗2 ̺b) = Tr ̺a̺b. For ρ = ̺ ̺ we can estimate Tr ̺ , modifications, when I Λ is not trace-preserving. In ⊗ m 2 ⊗ which gives us an estimate of i=1 λi , where λi are the this case experimental implementations require postse- eigenvalues of ̺. N.B. Tr ̺2 is real hence there is no need lections, which result ingn′ = nTr (I Λ(̺) copies of nor- to sweep the phase ϕ in the interferometer,P it can be fixed ⊗ malized states I Λ(̺)/Tr (I Λ(̺)). The spectrum es- at ϕ = 0. timation procedure⊗ is not affected,⊗ however,g before check- In general, in order to calculate the spectrum of any ing the conditiong (4) the lowestg eigenvalue has to be m m density matrix ̺ we need to estimate m 1 × 2 3 m − rescaled by the factor Tr (I Λ(̺)). parameters Tr̺ , Tr̺ ,... Tr̺ . For this we need the ⊗ controlled-shift operation. Given k systems of dimension Let us summarize our findings. Given n copies of a bi- m we define the shift V (k) as partite d d system describedg by some unknown density operator ⊗̺ we can test for entanglement either by esti- (k) V φ1 φ2 ... φk = φk φ1 ... φk 1 , (11) mating ̺ and applying criterion (2), or, more directly, − by performing the measurements we have just described. for any states φ . Such an operation can be easily con- The state estimation involves estimating d4 1 param- structed by cascading k 1 swaps V . This time, if we − k − eters of ̺, most of which are of no relevance for the en- prepare ρ = ̺⊗ the interference will be modified by the tanglement detection. The optimal state estimations rely factor on joint measurements on all copies of ̺. However, one m can construct less efficient but simpler state estimation k (k) k k Tr ̺⊗ V = Tr ̺ = λi . (12) methods which involve measurements only on individual i=1 X copies. Our more direct, interferometry based, method (k) requires estimations of only d2 1 parameters and joint Thus measuring the average values of V for k = − 2, 3...m gives us effectively the spectrum of ̺. In par- operations on d copies of ̺′. The most demanding, from ticular, in our case, we obtain the spectrum (and the the experimental point of view, is our second method. It

3 is a measurement with an outcome which is an estimate [1] Erwin Schr¨odinger. Naturwiss., 48:807, 1935. of just one parameter, but, like the optimal state esti- [2] A. Einstein, B. Podolsky, and N. Rosen. Physical Review, mation, the measurement involves joint operations on all 47:777, 1935. [3] J.S. Bell. Physics, 1:195, 1964. copies of ̺′. Both direct and indirect entanglement detec- tions have their own merits. Depending on the context, [4] A. K. Ekert. Quantum cryptography based on bell theo- Physical Review Letters applications, and technologies involved one can choose rem. , 67(6):661–663, 1991. [5] J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. one or the other. Heinzen. Optimal frequency measurements with maxi- Direct entanglement detections, can be employed as mally correlated states. Physical Review A, 54(6):R4649– sub-routines in quantum computation. For example, one R4652, 1996. may consider performing or not performing a quantum [6] V. Giovannetti, S. Lloyd, and L. Maccone. Quantum- operation on a given quantum system conditioned on enhanced positioning and clock synchronization. Nature, some part of quantum data being entangled or not. In 412(6845):417–419, 2001. fact direct entanglement detections can be viewed as [7] M. Horodecki, P. Horodecki, and R. Horodecki. Separa- quantum computations solving an inherently quantum bility of mixed states: Necessary and sufficient condi- decision problem: given as an input n copies of ̺ de- tions. Physics Letters A, 223(1-2):1–8, 1996. cide whether ̺ is entangled. Here the input data is [8] G. Alber, T. Beth, M. Horodecki, P. Horodecki, quantum and such a decision problem cannot even be R. Horodecki, M. R¨ottler, H. Weinfurter, R. F. Werner, and A. Zeilinger. Quantum information: an introduction even formulated for classical computers. Nonetheless the to basic theoretical concepts and experiments., volume 173 problem is perfectly well defined for quantum comput- of Springer Tracts in Modern Physics. Springer, 2001. ers. Finally, let us add that the method presented here [9] R. F. Werner. Quantum states with einstein-podolsky- can be easily generalized to cover all linear maps tests rosen correlations admitting a hidden-variable model. for arbitrary multiparticle entanglement [18] and the so Physical Review A, 40(8):4277–4281, 1989. called k-positive map tests detecting Schmidt numbers [10] A. Peres. Separability criterion for density matrices. of density matrices [19]. Modification of the method to Physical Review Letters, 77(8):1413–1415, 1996. the case of two distant labs (i. e. under restrictions to [11] P. Horodecki. From limits of nonlinear quantum opera- local operations and classical communication [8]) will be tions to multicopy entanglement witnesses and spectrum considered elsewhere. states estimation. quant-ph/0111036. For the sake of completeness we should also mention [12] F. Verstraete, K. Audenaert, J. Dehaene, and D. de Moor. A comparison of the entanglement measures here that there are two-particle observables, called en- negativity and concurence. quant-ph/0108021. tanglement witnesses which can detect quantum entan- [13] S. Pancharatnam. Generalized theory of interference and glement is some special cases (see [20,8]). They have its applications. Proceedings of the Indian Academy of positive mean values on all separable states and nega- Science, XLIV(5):247–262, 1956. tive on some entangled states. Therefore any individual [14] E. Sj¨oqvist, A. K. Pati, A. Ekert, J. S. Anandan, M. Er- entanglement witness leaves many entangled states un- icsson, D. K. L. Oi, and V. Vedral. Geometric phases for detected. When ̺ is unknown we need to check infinitely mixed states in interferometry. Physical Review Letters, many witnesses, which effectively reduces this approach 85(14):2845–2849, 2000. to the quantum state estimation. However, let us point [15] M. Keyl and R. F. Werner. Estimating the spectrum of out any witness defines a positive map which can be used a density operator. Physical Review A, 64 (5):052311-1– in our test. 052311-5, 2001. [16] J. I. Cirac, A. K. Ekert, and C. Macchiavello. Opti- To conclude, we have demonstrated that direct and mal purification of single qubits. Physical Review Letters, physically implementable methods of entanglement de- 82(21):4344–4347, 1999. tection are possible. They are equivalent to the most [17] A. Barenco, A. Berthiaume, D. Deutsch, A. Ekert, powerful mathematical separability criteria known to R. Jozsa, and C. Macchiavello. Stabilization of quan- date [7]. We have described their possible realizations tum computations by symmetrization. SIAM Journal on in the generic terms of quantum gates and networks. Computing, 26(5):1541–1557, 1997. These generic components can be implemented using sev- [18] M. Horodecki, P. Horodecki, and R. Horodecki. Separa- eral experimental techniques ranging from trapped ions bility of n-particle mixed states: necessary and sufficient to quantum dots [21]. conditions in terms of linear maps. Physics Letters A, This research was supported in part by the Polish 283(1-2):1–7, 2001. Committee for Scientific Research, European Commis- [19] B. M. Terhal, P. Horodecki. Schmidt number for density matrices. Physical Review A, 61 ():040301-040305, 2000. sion, Elsag Spa, EPSRC, and the Royal Society, London. [20] D. Bruß, J. I. Cirac, P. Horodecki, F. Hulpke, B. Kraus, M. Lewenstein, A. Sanpera. Reflections upon separability and distillability. quant-ph/0110081. [21] Dirk Bouwmeester, Artur K. Ekert, and Anton Zeilinger. The physics of quantum information : quantum cryp- tography, quantum teleportation, quantum computation.

4 Springer, New York, 2000. Dirk Bouwmeester, Artur K. Ekert, Anton Zeilinger (eds.).

5