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

[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 物理学报 65 170301]
[10]	Ren B C, Deng F G 2015 Acta Phys. Sin. 64 160303 (in Chinese) [任宝藏, 邓富国 2015 物理学报 64 160303]
[11]	Zong X L, Yang M 2016 Acta Phys. Sin. 65 080303 (in Chinese) [宗晓岚, 杨名 2016 物理学报 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 物理学报 65 170301]
[10]	Ren B C, Deng F G 2015 Acta Phys. Sin. 64 160303 (in Chinese) [任宝藏, 邓富国 2015 物理学报 64 160303]
[11]	Zong X L, Yang M 2016 Acta Phys. Sin. 65 080303 (in Chinese) [宗晓岚, 杨名 2016 物理学报 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

[1]	陶志炜, 任益充, 艾则孜姑丽·阿不都克热木, 刘世韦, 饶瑞中. 基于纠缠相干态的量子照明雷达. 物理学报, 2021, 70(17): 170601. doi: 10.7498/aps.70.20210462
[2]	宋悦, 李军奇, 梁九卿. 级联环境下三量子比特量子关联动力学研究. 物理学报, 2021, 70(10): 100301. doi: 10.7498/aps.70.20202133
[3]	田宇玲, 冯田峰, 周晓祺. 基于冗余图态的多人协作量子计算. 物理学报, 2019, 68(11): 110302. doi: 10.7498/aps.68.20190142
[4]	梁建武, 程资, 石金晶, 郭迎. 基于量子图态的量子秘密共享. 物理学报, 2016, 65(16): 160301. doi: 10.7498/aps.65.160301
[5]	丁东, 何英秋, 闫凤利, 高亭. 六光子超纠缠态制备方案. 物理学报, 2015, 64(16): 160301. doi: 10.7498/aps.64.160301
[6]	吴向艳, 徐艳玲, 於亚飞, 张智明. 利用非稳定子态容错实现密集旋转操作. 物理学报, 2014, 63(22): 220304. doi: 10.7498/aps.63.220304
[7]	李洪伟, 周云龙, 刘旭, 孙斌. 基于随机子空间结合稳定图的气液两相流型分析. 物理学报, 2012, 61(3): 030508. doi: 10.7498/aps.61.030508
[8]	曲照军, 马晓光, 徐秀玮, 杨传路. 可控三模纠缠相干态的产生. 物理学报, 2012, 61(3): 034206. doi: 10.7498/aps.61.034206
[9]	徐健, 陈小余, 李海涛. 多进制量子图态纠缠的确定. 物理学报, 2012, 61(22): 220304. doi: 10.7498/aps.61.220304
[10]	肖芳英, 陈汉武. 量子稳定子码的差错纠正与译码网络构建. 物理学报, 2011, 60(8): 080303. doi: 10.7498/aps.60.080303
[11]	李体俊. 纠缠态投影算符的积分. 物理学报, 2009, 58(6): 3665-3669. doi: 10.7498/aps.58.3665
[12]	唐有良, 刘翔, 张小伟, 唐筱芳. 用一个纠缠态实现多粒子纠缠态的量子隐形传送. 物理学报, 2008, 57(12): 7447-7451. doi: 10.7498/aps.57.7447
[13]	夏云杰, 高德营. 纠缠相干态及其非经典特性. 物理学报, 2007, 56(7): 3703-3708. doi: 10.7498/aps.56.3703
[14]	李艳玲, 冯健, 於亚飞. 量子纠缠态的普适远程克隆. 物理学报, 2007, 56(12): 6797-6802. doi: 10.7498/aps.56.6797
[15]	刘传龙, 郑亦庄. 纠缠相干态的量子隐形传态. 物理学报, 2006, 55(12): 6222-6228. doi: 10.7498/aps.55.6222
[16]	姜金龙, 李文杰, 周立, 赵汝光, 杨威生. 高指数稳定硅表面的低能电子衍射图分析. 物理学报, 2003, 52(1): 156-162. doi: 10.7498/aps.52.156
[17]	郑亦庄, 戴玲玉, 郭光灿. 三粒子纠缠Ｗ态的隐形传态. 物理学报, 2003, 52(11): 2678-2682. doi: 10.7498/aps.52.2678
[18]	陶孟仙, 路洪, 佘卫龙. 增加光子纠缠相干态的统计性质. 物理学报, 2002, 51(9): 1996-2001. doi: 10.7498/aps.51.1996
[19]	石名俊, 杜江峰, 朱栋培. 量子纯态的纠缠度. 物理学报, 2000, 49(5): 825-829. doi: 10.7498/aps.49.825
[20]	石名俊, 杜江峰, 朱栋培, 阮图南. 混合纠缠态的几何描述. 物理学报, 2000, 49(10): 1912-1918. doi: 10.7498/aps.49.1912

计量

文章访问数: 8123
PDF下载量: 231
被引次数: 0

姓名
邮箱
手机号码
标题
留言内容
验证码

搜索

留言板