量子系统的纠缠探测与纠缠测量

摘要/Abstract

摘要：

在回顾了量子系统中有关纠缠探测的各种方法和纠缠度不同定义的基础上，总结了探测两体或多体纠缠的几种分离判据；分析了它们与正映射之间及其相互之间的关系；重点分析了纠缠目击者这种特殊的分离判据，包括它的定义以及构造方法，并且从实验观点分析了它的应用。在已提出的公理化假设的纠缠测量思想的基础上，讨论了理论上各种两体纠缠度和多体纠缠度的定义并且讨论了各种纠缠度在实验中的估计问题。最后对各种非线性分离判据进行了分析。

关键词: 量子信息, 纠缠探测, 纠缠度, 纠缠目击者, 分离判据, Legendre变换

Abstract:

Various kinds of methods of entanglement detection and definitions of entanglement measures in quantum systems were reviewed. For entanglement of two or more particles, some separability criterions were summarized. Their relationships with positive map and the relationships between themselves were also analyzed. Particularly, concentrations on entanglement witness are paid which is a special criterion involving its definition and construction methods. Further, its applications are also investigated from the viewpoint of experiment. Based on axiomatic hypothesis of entanglement measure, the theoretical definitions of two-particle and multipartite entanglement, and their estimations in experiment were discussed. At last, various nonlinear separability criterions were analyzed.

Key words: quantum information, entanglement detection, entanglement measure, entanglement witness, separability criterion, Legendre transform

中图分类号:

O431.2

杨霏丛爽. 量子系统的纠缠探测与纠缠测量[J]. J4, 2011, 28(4): 391-401.

YANG Fei, CONG Shuang. Entanglement detection and measure of quantum systems[J]. J4, 2011, 28(4): 391-401.

参考文献

[1] Einstein A, Podolsky B, Rosen N. Can quantum-mechanical description of physical reality be considered complete? [J]. Phys. Rev., 1935, 47(10): 777-780.
[2] Schrödinger E. Die gegenwärtige Situation in der Quantenmechanik [J]. Die Naturwissenschaften 23 (1935) 807_812; 823_828; 844_849.
[3] Bell J S. 1964, Physics (Long Island City, N.Y.) 1, 195.
[4] Bennett C H, Brassard G, Crepeau C, Jozsa R, Peres A, and Wootters W K. Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels [J]. Phys. Rev. Lett., 1993, 70 (13): 1895-1899
[5] Ekert A K. Quantum cryptography based on Bell’s theorem [J]. Phys. Rev. Lett.,1991, 67 (6): 661-663
[6] Raussendorf R, Briegel H J. A one-way quantum computer [J]. Phys Rev. Lett., 2001, 86 (22):5188-5191.
[7] Leibfried D, Knill E, et al. Creation of a six-atom ‘Schrödinger cat’ state [J]. Nature, 2005, 438: 639-642.
[8] Häffner H, et al. Scalable multiparticle entanglement of trapped ions [J]. Nature, 2005, 438: 643-646.
[9] Lu C Y, et al. Experimental entanglement of six photons in graph states [J]. Nature Phys., 2007, 3: 91-95.
[10] Gao W B, et al. Experimental demonstration of a hyper-entangled ten-qubit Schrödinger cat state [J]. Nature Physics,2010, 6, 331-335
[11] Neumann P, et al. Multipartite entanglement among single spins in diamond [J]. Science, 2008, 320 (5881): 1326-1329.
[12] Hald J, et al. Spin squeezed atoms: A macroscopic entangled ensemble created by light [J]. Phys. Rev. Lett., 1999, 83 (77): 1319-1322.
[13] Mandel O, et al. Controlled collisions for multi-particle entanglement of optically trapped atoms [J]. Nature, 2003, 425: 937.
[14] Peres A. Collective tests for quantum nonlocality [J]. Phys. Rev. A, 1996, 54(4): 2685-2689.
[15] Jamiolkowski A. Linear transformations which preserve trace and positive semidefiniteness of operators [J]. Rep. Math. Phys., 1972 3: 275-278
[16] Horodecki M, Horodecki P, and Horodecki R. Inseparable Two Spin- 1/2 density matrices can be distilled to a singlet form [J]. Phys. Rev. Lett., 1997, 78 (4): 574-577
[17] Terhal B M. A family of indecomposable positive linear maps based on entangled quantum states [J]. Linear Algebra Appl. 323, 61-73
[18] Horodecki M, Oppenheim J, and Horodecki R. Are the laws of entanglement theory thermodynamical [J]. Phys. Rev. Lett., 2002, 89, 240403
[19] Vedral V, and Plenio M B. Entanglement measures and purification procedures [J]. Phys. Rev. A, 1998, 57 (3): 1619-1633
[20] Bennett C H, DiVincenzo D P, Smolin J, and Wootters W K. Mixed-state entanglement and quantum error correction [J]. Phys. Rev. A, 1996, 54(5): 3824-3851
[21] Eisert J, and Plenio M B. A comparison of entanglement measure [J] J. Mod. Opt., 1999, 46 (1): 145-154
[22] Hall W. A new criterion for indecomposability of positive maps [J]. J. Phys. A: Math. Gen. 39 (2006) 14119.
[23] Hiroshima T. Majorization criterion for distillability of a bipartite quantum state [J]. Phys. Rev. Lett., 2003, 91, 057902.
[24] Zhang Y D. Principles of Quantum Information Physics (量子信息物理基本原理) [M] Science Press (Beijing) 2005, 81-85 (in Chinese).
[25] Augusiak R, and Stasińska J. General scheme for construction of scalar separability criteria from positive maps [J]. Phys. Rev. A, 2008, 77, 010303(R)
[26] Uffink J. Quadratic Bell inequalities as tests for multipartite entanglement [J]. Phys. Rev. Lett., 2002, 88, 230406.
[27] Horodecki M, Horodecki P, Horodecki R. Separability of n-particle mixed states: necessary and sufficient conditions in terms of linear maps [J]. Phys. Lett. A, 2001, 283 (1-2): 1-7
[28] Yu C S, Song H S. Separability criterion of tripartite qubit systems [J]. Phys. Rev. A, 2005, 72, 022333
[29] Horodecki M, Horodecki P, Horodecki R. Separability of mixed states: necessary and sufficient conditions [J]. Phys. Lett. A, 1996, 223 (1-2): 1-8
[30] Gühne O and Hyllus P. Investigating Three Qubit Entanglement with Local measurement [J]. International Journal of Theoretical Physics, 2003, 42(5):1001-1013
[31] Gühne O, Hyllus P, and Bruss D. et al. Experimental detection of entanglement via witness operators and local measurements [J]. Journal of Modern Optics, 2003, 50 (6-7): 1079-1102.
[32] Gühne O and Géza Tóth. Entanglement detection [J]. Physics Reports, 2009, 474 (1-6): 1-75.
[33] Bennett C H, Brassard G, Popescu S, et al. Purification of noisy entanglement and faithful teleportation via noisy channels [J]. Phys. Rev. Lett., 1997, 78 (10):2031
[34] Rains, E., 2000, Semi-definite program for distillable entanglement, eprint quant-ph/0008047.
[35] Vedral V, and. Plenio M B. Entanglement measures and purification procedures [J]. Phys. Rev. A, 1998, 57 (3): 1619-1633.
[36] Bennett C H, DiVincenzo D P, Smolin J, and Wootters W K. Mixed-state entanglement and quantum error correction [J]. Phys. Rev. A, 1996, 54(5): 3824-3851
[37] Hill S, Wootters W K. Entanglement of a pair of quantum bits. [J]. Phys. Rev. Lett., 1997, 78(26):5022-5025
[38] Vidal G and Werner R F. Computable measure of entanglement [J]. Phys. Rev. A, 2002, 65, 032314
[39] Vidal G and Tarrach R. Robustness of entanglement [J]. Phys. Rev. A, 1999, 59 (1): 141-155.
[40] Rudolph O. A new class of entanglement measures [J]. J. Math. Phys., 2001, 42 (11): 5306-5314.
[41] Tucci R. Entanglement of distillation and conditional mutual information, eprint quant-ph/0202144.
[42] Karnas S and Lewenstein M. Separable approximations of density matrices of composite quantum systems [J]. J. Phys. A: Math. Gen., 2001, 34, 6919.
[43] Brandão F G S L, and Vianna R O. Separable multipartite mixed states- operational asymptotically necessary and sufficient conditions [J]. Phys. Rev. Lett., 2004, 93, 220503.
[44] Yang T, Zhang Q, Zhang J, et al. All-versus-nothing violation of local realism by two-photon, four-dimensional entanglement [J]. Phys. Rev. Lett., 2005, 95, 240406.
[45] Horodecki R et al. Quantum entanglement [J]. Rev. Mod. Phys., 2009, 81(2): 865-942.
[46] Gühne O， Reimpell M and Werner R F. Estimating entanglement measures in experiments [J]. Phys. Rev. Lett., 2007, 98, 110502
[47] Eisert J, Brandão F G S L and Audenaert K M R. Quantitative entanglement witnesses [J]. New Journal of Physics, 2007, 9, 46
[48] Gühne O, and Lütkenhaus N. Nonlinear entanglement witness [J]. Phys. Rev. Lett., 2006, 96,170502.
[49] Tóth G, Knapp C, Gühne O, Briegel H J. Optimal spin squeezing inequalities detect bound entanglement in spin models [J]. Phys. Rev. Lett., 2007, 99, 250405.