Abstract
In this review, we introduce well-known Bell inequalities, the relations between the Bell inequality and quantum separability, and the entanglement distillation of quantum states. It is shown that any pure entangled quantum state violates one of Bell-like inequalities. Moreover, quantum states that violate any one of these Bell-like inequalities are shown to be distillable. New Bell inequalities that detect more entangled mixed states are also introduced.
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Einstein A, Podolsky B, Rosen N. Can quantum-mechanical description of physical reality be considered complete? Phys Rev, 1935, 47: 777–780
Bell J S. On the Einstein-Podolsky-Rosen Paradox. Physics, 1964, 1: 195–200
Brukner Č, Żukowski M, Zeilinger A. Quantum communication complexity protocol with two entangled qutrits. Phys Rev Lett, 2002, 89: 197901
Scarani V, Gisin N. Quantum communication between N partners and Bell’s inequalities. Phys Rev Lett, 2001, 87: 117901
Acín A, Gisin N, Scarani V. Security bounds in quantum cryptography using d-level systems. Quantum Inf Comput, 2003, 3: 563–580
Clauser J F, Horne M A, Shimony A, et al. Proposed experiment to test local hidden-variable theories. Phys Rev Lett, 1969, 23: 880–884
Mermin N D. Extreme quantum entanglement in a superposition of macroscopically distinct states. Phys Rev Lett, 1990, 65: 1838–1840
Ardehali M. Bell inequalities with a magnitude of violation that grows exponentially with the number of particles. Phys Rev A, 1992, 46: 5375–5378
Belinskii A V, Klyshko D N. Interference of light and Bell’s theorem. Phys Usp, 1993, 36: 653–693
Werner R F, Wolf M M. All-multipartite Bell-correlation inequalities for two dichotomic observables per site. Phys Rev A, 2001, 64: 032112
Żukowski M, Brukner Č. Bells Theorem for general N-qubit states. Phys Rev Lett, 2002, 88: 210401
Żukowski M, Brukner Č, Laskowski W, et al. Do all pure entangled states violate Bells inequalities for correlation functions? Phys Rev Lett, 2002, 88: 210402
Chen K, Albeverio S, Fei S M. Two-setting Bell inequalities for many qubits. Phys Rev A, 2006, 74: 050101 (R)
Wu Y C, Badziag P, Wiesniak M, et al. Extending Bell inequalities to more parties. Phys Rev A, 2008, 77: 032105
Collins D, Gisin N, Linden N, et al. Bell inequalities for arbitrarily high-dimensional systems. Phys Rev Lett. 2002, 88: 040404
Gisin N. Bell’s inequality holds for all non-product states. Phys Lett A, 1991, 154: 201–202
Gisin N, Peres A. Maximal violation of Bell’s inequality for arbitrarily large spin. Phys Lett A, 1992, 162: 15–17
Chen J L, Wu C F, Kwek L C, et al. Gisin’s Theorem for three qubits. Phys Rev Lett, 2004, 93: 140407
Ekert A K. Quantum cryptography based on Bells theorem. Phys Rev Lett, 1991, 67: 661–663
Acín A, Brunner N, Gisin N, et al. Device-independent security of quantum cryptography against collective attacks. Phys Rev Lett, 2007, 98: 230501
Horodecki R, Horodecki M, Horodecki P. Teleportation, Bell’s inequalities and inseparability. Phys Lett A, 1996, 222: 21–25
Lee S, Joo J, Kim J. Teleportation capability, distillability, and nonlocality on three-qubit states. Phys Rev A, 2007, 76: 012311
Dür W. Multipartite bound entangled states that violate Bells inequality. Phys Rev Lett, 2001, 87: 230402
Acín A. Distillability, Bell inequalities, and multiparticle bound entanglement. Phys Rev Lett, 2002, 88: 027901
Acín A, Scarani V, Wolf M M. Bells inequalities and distillability in N-quantum-bit systems. Phys Rev A, 2002, 66: 042323
Braunstein S L, Mann A, Revzen M. Maximal violation of Bell inequalities for mixed states. Phys Rev Lett, 1992, 68: 3259–3261
Horodecki R, Horodecki P, Horodecki M. Violating Bell inequality by mixed spin-12 states: Necessary and sufficient condition. Phys Lett A, 1995, 200: 340–344
Pan J W, Bouwmeester D, Daniell M, et al. Experimental test of quantum nonlocality in three-photon Greenberger-Horne-Zeilinger entanglement. Nature, 2000, 403: 515–519
Sun B Z, Fei S M. Bell inequalities classifying biseparable three-qubit states. Phys Rev A, 2006, 74: 032335
Chen J L, Deng D L, Hu M G. Gisins theorem for two d-dimensional systems based on the Collins-Gisin-Linden-Masser-Popescu inequality. Phys Rev A, 2008, 77: 060306 (R)
Fu L B. General correlation functions of the Clauser-Horne-Shimony-Holt inequality for arbitrarily high-dimensional systems. Phys Rev Lett, 2004, 92: 130404
Chen J L, Deng D L. Tight correlation-function Bell inequality for multipartite d-dimensional systems. Phys Rev A, 2009, 79: 012111
Gisin N. Bell inequality for arbitrary many settings of the analyzers. Phys Lett A, 1999, 260: 1–3
Ji S W, Lee J Y, Lim J, et al. Multisetting Bell inequality for qudits. Phys Rev A, 2008, 78: 052103
Son W, Lee J, Kim M S. Generic Bell inequalities for multipartite arbitrary dimensional systems. Phys Rev Lett, 2006, 96: 060406
Li M, Fei S M. Gisins Theorem for arbitrary dimensional multipartite states. Phys Rev Lett, 2010, 104: 240502
Ou Y C, Fan H, Fei S M. Proper monogamy inequality for arbitrary pure quantum states. Phys Rev A, 2008, 78: 012311
Li M, Fei S M, Li-Jost X Q. Bipartite Bell inequality and maximal violation. Commun Theor Phys, 2011, 55: 418–420
Uhlmann A. Fidelity and concurrence of conjugated states. Phys Rev A, 2000, 62: 032307
Rungta P, Buzek V, Caves C M, et al. Universal state inversion and concurrence in arbitrary dimensions. Phys Rev A, 2001, 64: 042315
Albeverio S, Fei S M. A note on invariants and entanglement. J Opt B: Quantum Semiclass Opt, 2001, 3: 223–227
Acin A, Andrianov A, Costa L, et al. Generalized Schmidt decomposition and classification of three-quantum-bit states. Phys Rev Lett, 2000, 85: 1560–1563
Horodecki M, Horodecki P, Horodecki R. Mixed-state entanglement and distillation: Is there a bound entanglement in nature? Phys Rev Lett, 1998, 80: 5239–5242
Peres A. Separability criterion for density matrices. Phys Rev Lett, 1996, 77: 1413–1415
Życzkowski K, Horodecki P. Volume of the set of separable states. Phys Rev A, 1998, 58: 883–892
Horodecki M, Horodecki P, Horodecki R. Separability of mixed states: necessary and sufficient conditions. Phys Lett A, 1996, 223: 1–8
Yu S X, Pan J W, Chen Z B, et al. Comprehensive test of entanglement for two-level systems via the indeterminacy relationship. Phys Rev Lett, 2003, 91: 217903
Cabello A. Proposal for revealing quantum nonlocality via local contextuality. Phys Rev Lett, 2010, 104: 220401
Amselem E, Rådmark M, Bourennane M, et al. State-independent quantum contextuality with single photons. Phys Rev Lett, 2009, 103: 160405
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Li, M., Fei, S. & Li-Jost, X. Bell inequality, separability and entanglement distillation. Chin. Sci. Bull. 56, 945–954 (2011). https://doi.org/10.1007/s11434-011-4395-1
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DOI: https://doi.org/10.1007/s11434-011-4395-1