构造纠缠目击的一般方法

杨莹; 曹怀信

doi:10.7498/aps.67.20172697

摘要

量子纠缠作为量子通信和量子计算过程中不可缺少的资源，在量子信息领域中有着广泛的应用.如何判定一个给定的量子态是否为纠缠态仍然是一个重要的课题.纠缠目击是一种特殊的自伴算子，它可以用来判断一个量子态是否为纠缠态.本文首先从纠缠目击的定义入手，给出构造纠缠目击的一般方法，证明了当一个可测量A在可分纯态上的最大期望CA严格小于它的最大特征值max（A）时，对任何满足条件CA C （A）的参数C，算子WC=CI-A都是一个纠缠目击；然后，作为应用得到了利用图态的稳定子构造纠缠目击的一系列方法.

关键词:

纠缠目击 /
图态 /
稳定子

Abstract

Quantum entanglement, as an indispensable resource in quantum communication and quantum computation, is widely used in the field of quantum information. However, people's understanding on entanglement is quite limited both theoretically and experimentally. How to determine whether a given quantum state is entangled is still an important task. The entanglement witness is a kind of special self-adjoint operator, it can be used to determine whether a quantum state is an entangled state. This provides a new direction for the determination of entangled states. Entanglement witness has its own unique characteristics in various kinds of entanglement criterion. It is the most effective tool for detecting multipartite entanglement, and the most useful method to detect entanglement in experiments. In the background of quantum theory, we use theory of operators to make a thorough and systematic study of the construction of entanglement witness in this paper. First, from the definition of an entanglement witness, a general method is given to construct an entanglement witness. It is proved that when the maximal expectation CA of an observable A in the separable pure states is strictly less than its biggest eigenvalue max(A), the operator WC=CI-A is an entanglement witness provided that CA C max(A). Although the entanglement witness WCA can detect more entangled states than WC, but it is difficult to calculate the exact value of CA, and the estimate of the upper bound of CA is easier. Therefore, it is more convenient to construct entanglement witness WC than WCA. In quantum computation, a graph state is a special kind of multi-qubit state that can be represented by a graph. Each qubit is represented by a vertex of the graph, and there is an edge between every interacting pair of qubits. Graph states play a crucial role in many applications of quantum information theory, such as quantum error correcting codes, measurement-based quantum computation, and quantum simulation. Consequently, a significant effort is devoted to the creation and investigation of graph states. In the last part of this paper, as applications of our method, a series of methods for constructing an entanglement witness is obtained in the stabilizer formalism. It is also proved that how entanglement witnesses can be derived for a given graph state, provided some stabilizing operators of the graph state are known. Especially, when A is made up of some stabilizing operators of a graph state, entanglement witness WCA becomes one in literature.

Keywords:

作者及机构信息

杨莹^1,2,
曹怀信¹

1.
陕西师范大学数学与信息科学学院, 西安 710119;

2.
运城学院数学与信息技术学院, 运城 044000

通信作者: 曹怀信, caohx@snnu.edu.cn

基金项目: 国家自然科学基金（批准号：11371012，11601300，11571213，11771009）和中央高校基本科研业务费专项资金（批准号：GK201703093）资助的课题.

Authors and contacts

Yang Ying^1,2,
Cao Huai-Xin¹

1.
School of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710119, China;

2.
School of Mathematics and Information Technology, Yuncheng College, Yuncheng 044000, China

Corresponding author: Cao Huai-Xin, caohx@snnu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11371012, 11601300, 11571213, 11771009) and the Fundamental Research Funds for the Central Universities of Ministry of Education of China (Grant No. GK201703093).

参考文献

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[2]	Ekert A K 1991 Phys. Rev. Lett. 67 661
[3]	Steane A 1998 Rep. Prog. Phys. 61 117
[4]	Mattle K, Weinfurter H, Kwiat P G, Zeilinger A 1996 Phys. Rev. Lett. 76 4656
[5]	Hillery M, Buvek V, Berthiaume A 1999 Phys. Rev. A 59 1829
[6]	Long G L, Liu X S 2002 Phys. Rev. A 65 032302
[7]	Sheng Y B, Zhou L 2017 Sci. Bull. 62 1025
[8]	Deng F G, Ren B C, Li X H 2017 Sci. Bull. 62 46
[9]	Cong M Y, Yang J, Huang Y X 2016 Acta Phys. Sin. 65 170301 (in Chinese) [丛美艳, 杨晶, 黄燕霞 2016 65 170301]
[10]	Ren B C, Deng F G 2015 Acta Phys. Sin. 64 160303 (in Chinese) [任宝藏, 邓富国 2015 64 160303]
[11]	Zong X L, Yang M 2016 Acta Phys. Sin. 65 080303 (in Chinese) [宗晓岚, 杨名 2016 65 080303]
[12]	Yang F, Cong S 2011 Chin. J. Quant. Elect. 28 391 (in Chinese) [杨霏, 丛爽 2011 量子电子学报 28 391]
[13]	Lewenstein M, Kraus B, Cirac J I, Horodecki P 2000 Phys. Rev. A 62 052310
[14]	Lewenstein M, Kraus B, Horodecki P, Cirac J I 2001 Phys. Rev. A 63 044304
[15]	Tth G, Ghne O 2005 Phys. Rev. Lett. 94 060501
[16]	Ghne O, Hyllus P, Bruss D, Ekert A, Lewenstein M, Macchiavello C, Sanpera A 2002 Phys. Rev. A 66 062305
[17]	Tth G 2004 Phys. Rev. A 69 052327
[18]	Brukner C, Vedral V, Zeilinger A 2006 Phys. Rev. A 73 012110
[19]	Wu L A, Bandyopadhyay S, Sarandy M S, Lidar D A 2005 Phys. Rev. A 72 032309
[20]	Tth G, Ghne O 2005 Phys. Rev. A 72 022340
[21]	Doherty A C, Parrilo P A, Spedalieri F M 2005 Phys. Rev. A 71 032333
[22]	Vianna R O, Doherty A C 2006 Phys. Rev. A 74 052306
[23]	Jafarizadeh M A, Rezaee M, Yagoobi S K A S 2005 Phys. Rev. A 72 062106
[24]	Jafarizadeh M A, Rezaee M, Ahadpour S 2006 Phys. Rev. A 74 042335
[25]	Jafarizadeh M A, Najarbashi G, Habibian H 2007 Phys. Rev. A 75 052326
[26]	Jafarizadeh M A, Sufiani R, Nami S, Golmohammadi M 2012 Quantum. Inf. Process. 11 729
[27]	Cheng S, Chen J, Wang L 2017 Physics 46 416 (in Chinese) [程嵩, 陈靖, 王磊 2017 物理 46 416]
[28]	Deng D L, Li X P, Sarma S D 2017 Phys. Rev. X 7 021021
[29]	Levine Y, Yakira D, Cohen N, Shashua A 2017 arXiv: 1704.01552
[30]	Carleo G, Troyer M 2017 Science 355 602
[31]	Gao X, Duan L M 2017 Nature Commun. 8 662
[32]	Tth G, Ghne O, Briegel H J 2005 Phys. Rev. Lett. 95 120405
[33]	Hein M, Eisert J, Briegel H J 2003 Phys. Rev. A 69 062311

施引文献

[1]	Bennett C H, Brassard G, Crpeau C, Jozsa R, Peres A, Wootters W K 1993 Phys. Rev. Lett. 70 1895
[2]	Ekert A K 1991 Phys. Rev. Lett. 67 661
[3]	Steane A 1998 Rep. Prog. Phys. 61 117
[4]	Mattle K, Weinfurter H, Kwiat P G, Zeilinger A 1996 Phys. Rev. Lett. 76 4656
[5]	Hillery M, Buvek V, Berthiaume A 1999 Phys. Rev. A 59 1829
[6]	Long G L, Liu X S 2002 Phys. Rev. A 65 032302
[7]	Sheng Y B, Zhou L 2017 Sci. Bull. 62 1025
[8]	Deng F G, Ren B C, Li X H 2017 Sci. Bull. 62 46
[9]	Cong M Y, Yang J, Huang Y X 2016 Acta Phys. Sin. 65 170301 (in Chinese) [丛美艳, 杨晶, 黄燕霞 2016 65 170301]
[10]	Ren B C, Deng F G 2015 Acta Phys. Sin. 64 160303 (in Chinese) [任宝藏, 邓富国 2015 64 160303]
[11]	Zong X L, Yang M 2016 Acta Phys. Sin. 65 080303 (in Chinese) [宗晓岚, 杨名 2016 65 080303]
[12]	Yang F, Cong S 2011 Chin. J. Quant. Elect. 28 391 (in Chinese) [杨霏, 丛爽 2011 量子电子学报 28 391]
[13]	Lewenstein M, Kraus B, Cirac J I, Horodecki P 2000 Phys. Rev. A 62 052310
[14]	Lewenstein M, Kraus B, Horodecki P, Cirac J I 2001 Phys. Rev. A 63 044304
[15]	Tth G, Ghne O 2005 Phys. Rev. Lett. 94 060501
[16]	Ghne O, Hyllus P, Bruss D, Ekert A, Lewenstein M, Macchiavello C, Sanpera A 2002 Phys. Rev. A 66 062305
[17]	Tth G 2004 Phys. Rev. A 69 052327
[18]	Brukner C, Vedral V, Zeilinger A 2006 Phys. Rev. A 73 012110
[19]	Wu L A, Bandyopadhyay S, Sarandy M S, Lidar D A 2005 Phys. Rev. A 72 032309
[20]	Tth G, Ghne O 2005 Phys. Rev. A 72 022340
[21]	Doherty A C, Parrilo P A, Spedalieri F M 2005 Phys. Rev. A 71 032333
[22]	Vianna R O, Doherty A C 2006 Phys. Rev. A 74 052306
[23]	Jafarizadeh M A, Rezaee M, Yagoobi S K A S 2005 Phys. Rev. A 72 062106
[24]	Jafarizadeh M A, Rezaee M, Ahadpour S 2006 Phys. Rev. A 74 042335
[25]	Jafarizadeh M A, Najarbashi G, Habibian H 2007 Phys. Rev. A 75 052326
[26]	Jafarizadeh M A, Sufiani R, Nami S, Golmohammadi M 2012 Quantum. Inf. Process. 11 729
[27]	Cheng S, Chen J, Wang L 2017 Physics 46 416 (in Chinese) [程嵩, 陈靖, 王磊 2017 物理 46 416]
[28]	Deng D L, Li X P, Sarma S D 2017 Phys. Rev. X 7 021021
[29]	Levine Y, Yakira D, Cohen N, Shashua A 2017 arXiv: 1704.01552
[30]	Carleo G, Troyer M 2017 Science 355 602
[31]	Gao X, Duan L M 2017 Nature Commun. 8 662
[32]	Tth G, Ghne O, Briegel H J 2005 Phys. Rev. Lett. 95 120405
[33]	Hein M, Eisert J, Briegel H J 2003 Phys. Rev. A 69 062311

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