Quantum entanglement Ryszard Horodecki 1 Pawe l Horodecki 3 Micha l Horodecki 1, Karol Horodecki 1,2 1 Institute of Theoretical Physics and Astrophysics University of Gda´nsk, 80–952 Gda´nsk, Poland 2 Faculty of Mathematics, Physics and Computer Science University of Gda´nsk, 80–952 Gda´nsk, Poland and 3 Faculty of Applied Physics and Mathematics, Technical University of Gda´nsk, 80–952 Gda´nsk, Poland All our former experience with application of quantum theory seems to say: what is predicted by quantum formalism must occur in laboratory. But the essence of quantum formalism — entangle- ment, recognized by Einstein, Podolsky, Rosen and Schr¨odinger — waited over 70 years to enter to laboratories as a new resource as real as energy. This holistic property of compound quantum systems, which involves nonclassical correlations be- tween subsystems, is a potential for many quantum processes, including “canonical” ones: quan- tum cryptography, quantum teleportation and dense coding. However, it appeared that this new resource is very complex and difficult to detect. Being usually fragile to environment, it is robust against conceptual and mathematical tools, the task of which is to decipher its rich structure. This article reviews basic aspects of entanglement including its characterization, detection, distil- lation and quantifying. In particular, the authors discuss various manifestations of entanglement via Bell inequalities, entropic inequalities, entanglement witnesses, quantum cryptography and point out some interrelations. They also discuss a basic role of entanglement in quantum com- munication within distant labs paradigm and stress some peculiarities such as irreversibility of entanglement manipulations including its extremal form — bound entanglement phenomenon. A basic role of entanglement witnesses in detection of entanglement is emphasized. Contents 5. Entanglement witnesses and Bell inequalities 26 6. Distinguished maps criteria: reduction I. Introduction 2 criterion and its extensions 26 7. Range criterion and its applications; PPT II. Entanglement as a quantum property of entanglement 26 compound systems 8 8. Matrix realignment criterion and linear contractions criteria 28 III. Pioneering effects based on entanglement 10 9. Some classes of important quantum states: A. Quantum key distribution based on entanglement 10 entanglement regions of parameters 29 B. Quantum dense coding 10 10. Characterization of bipartite separability in C. Quantum teleportation 11 terms of biconcurrence 29 D. Entanglement swapping 12 11. Enhancing separability criteria by local filters 30 E. Beating classical communication complexity bounds with entanglement 12 VII. Multipartite entanglement — similarities and differences 30 IV. Correlation manifestations of entanglement: A. Notion of full (m-partite) separability 30 Bell inequalities. 13 B. Partial separability 32 A. Bell theorem: CHSH inequality. 13 B. The optimal CHSH inequality for 2 2 systems 14 VIII. Further improvements of entanglement tests: × C. Nonlocality of quantum states and LHV model 14 nonlinear separability criteria 33 1. Pure states 14 A. Uncertainty relation based separability tests 33 2. Mixed states 14 B. Nonlinear improvement of entanglement witnesses 34 arXiv:quant-ph/0702225v2 20 Apr 2007 D. Bell theorem beyond CHSH-setting 15 C. Detecting entanglement with collective E. Logical versions of Bell’s theorem 16 measurements 35 F. Violation of Bell inequalities: general remarks 17 1. Physical implementations of entanglement criteria with collective measurements 35 V. Entropic manifestations of entanglement 18 2. Collective entanglement witnesses 36 A. Entropic inequalities: classical versus quantum 3. Detection of quantum entanglement as order 18 quantum computing with quantum data B. 1-entropic inequalities and negativity of structure 37 information 19 C. Majorization relations 20 IX. Classical algorithms detecting entanglement 37 VI. Bipartite entanglement 20 X. Quantum entanglement and geometry 38 A. Definition and basic properties 20 B. Main separability/entanglement criteria in XI. The paradigm of local operations and classical bipartite case 21 communication (LOCC) 39 1. Positive partial transpose (PPT) criterion 21 A. Quantum channel — the main notion 39 2. Separability via positive, but not completely B. LOCC operations 39 positive maps 21 3. Separability via entanglement witnesses 22 XII. Distillation and bound entanglement 41 4. Witnesses and experimental detection of A. One-way hashing distillation protocol 41 entanglement 24 B. Two-way recurrence distillation protocol 42 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).