多体复合量子态基于可观测量算子的纠缠测度

杨丽丽，闫栋华，王银珠

多体复合量子态基于可观测量算子的纠缠测度

杨丽丽，闫栋华，*王银珠

（太原科技大学应用科学学院，山西，太原 030024）

近些年，人们对量子纠缠的量化已经有了许多深入的研究，但是，许多已有的纠缠度量还是难以计算的。LuoShunlong基于互信息提出了两体量子态的可观测量关联，并得到了两体量子态的一类纠缠测度。本文将两体系统的纠缠测度推广到多体复合量子系统，并证明了其满足纠缠测度的必要物理条件。

多体量子系统；量子态；可观测量关联；互信息；纠缠测度

0 引言

量子纠缠作为量子系统之间的一种非经典关联，越来越受到人们的关注。量化纠缠的目的最初来自量子通信[1-2]。近年来，纠缠被认为是量子信息理论中的一种资源，并被广泛应用于量子通信和信息处理任务中[3-4]。目前已经有许多比较成熟的纠缠测度，比如Concurrence纠缠测度[5-10]，形成纠缠测度[11-12]，Negativity 纠缠测度[13]等。一般来说，纠缠度量应该满足以下几个公理[14]：

产生的联合概率分布为：

Luo Shunlong基于互信息提出了两体量子态的可观测量关联，并得到了两体量子态的一类纠缠测度[16]。本文将两体系统的纠缠测度推广到多体复合量子系统，并证明了其满足纠缠测度的必要物理性质。

1 主要结果

为了证明引理2，引入以下引理3。

引理2证明 先证必要性成立。

等号左边可化简为

同时等号右边可化简为

下面证明充分性成立。

性质1 （纠缠测度的非负性）

证明 由引理1有

性质2 （纠缠测度的局部酉不变性）

而对混合态来说,根据凸组合的结构,显然成立。

性质3 （纠缠测度的LOCC单调性）

证明 根据文献[20]，有

因此

所以

2 结论

本研究，定义了多体复合量子态基于可观测量算子的纠缠测度，并证明了该纠缠测度满足非负性，局部酉不变性，LOCC单调性等性质。

[1] Bennett C H, Brassard G, Popescu S, et al. Purification of noisy entanglement and faithful teleportation via noisy channels[J]. Phys. Rev. Lett,1996,78:2031 .

[2] Bennett C H, Divincenzo D P ,Smolin J A ,et al. Mixed State Entanglement and Quantum Error Correction[J]. Physical Review A :Atomic Molecular and Optical Physics,1996, 54(5):3824.

[3] Bennett C H, Wiesner S J. Communication via one- and two-particle operators on Einstein–Podolsky–Rosen states[J]. Phys. Rev. Lett, 1992,69:2881-2884.

[4] Bennett C H, Brassard G, Crepeau C, et al. Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels[J]. Phys. Rev. Lett,1993,70:1895-1899.

[5] Zhao H, Zhang M M, Fei S M, et al. Projection based lower bounds of concurrence for multipartite quantum systems[J].International Journal of Theoretical Physics,2020,59(6):1688-1698.

[6] Zhu X N, Li M, Zhang M M, et al. Lower bounds of concurrence for multipaitite state[J]. American Institute of Physics, 2012, 77:1424.

[7] Zhang L M, Gao T, Yan F L. Relations among k-ME concurrence, negativity, polynomial invariants, and tangle[J].Quantum Information Processing,2019,18(6):2223-2228.

[8] Qi X F, Gao T, Yan F L.Lower bounds of concurrence for N-qubit systems and the detection of k-nonseparability of multipartite quantum systems[J].Quantum Information Processing,2017,16(1):23.

[9] Li X S, Gao X H , Fei S M . Lower bound of concurrence based on positive maps[J]. Physical Review A,2011, 83(3):034303.

[10] Qi X F, Gao T, Yan F L.Measuring coherence with entanglement concurrence[J].Journal of Physics A: Mathematical and Theoretical,2017,50(28):285301.

[11] Chen K, Albeverio S, Fei S M. Entanglement of formation of bipartite quntum states[J]. Phys. Rev. Lett., 2005,95:210-501.

[12] Wang Y Z, Wang D X. Entanglement of Formation for Multipartite k-NonseparableStates[J].International Journalof Theoretical Physics,2016,55(1):517-525.

[13] Soojoon L, Chi D P, Oh S D, et al. Convex-roof extended negativity as an entanglement measure for bipartite quantum systems[J]. Phys. Rev. A,2003,68:062304.

[14] Vedral V, Plenio M B , Rippin M A ,et al. Quantifying Entanglement[J]. Physical Review Letters,1997, 78(12):2275-2279.

[15] Werner R F. Quantum states with Einstein - Podolsky -Rosen correletions admitting a hidden variable model[J].Phys. Rev.A,1989,40:4227-4281.

[16] Luo S L. Entanglement measures based on observable correlations[J]. Theoretical and Mathematical Physics, 2008, 155(3):896-904.

[17] Samuel R. Hedemann. Correlation and discordance: computable measures of nonlocal correlation[J]. Quantum information processing, 2020,19:189.

[18] Cover T M, Thomas J A. Elements of Information Theory[M]. New York: Wiley, 2006.

[19] SaiToh, Akira. Decoherence Suppression in Quantum Systems [M]. Singapore: World Scientific, 2009.

[20] Guo Y, Hou J C, Wang Y C. Concurrence for infinite- dimensional quantum systems[J]. Quantum Information Processing,2013, 12(8):2641-2653.

[21] Cover T M, Thomas J A. Elements of Information Theory[M].New York:Wiley,1991.

THE ENTANGLEMENT MEASURE OF THE MULTIPARTITE COMPOSITE QUANTUM STATES BASED ON OBSERVABLE OPERATOR

YANG Li-li，YAN Dong-hua，*WANG Yin-zhu

（School of Applied Sciences, Taiyuan University of Science and Technology, Taiyuan, Shanxi 030024, China）

In recent years, there are many research results on the quantization of quantum entanglement, but many of the existing entanglement measures are still difficult to calculate. In the paper “Entanglement measures based on observable correlations”, LUO Shun-long proposed an observable correlation measure of bipartite quantum states based on mutual information, and obtained a class of entanglement measures of bipartite quantum states. In this paper, we generalize the entanglement measure of the bipartite system to the multipartite composite quantum system, and prove that it satisfies the necessary physical conditions of entanglement measure.

multipartite quantum system; quantum states; observable correlations; mutual information; entanglement measure

1674-8085（2022）03-0001-05

O413.1

A

10.3969/j.issn.1674-8085.2022.03.001

2021-11-29；

2022-01-12

国家自然科学基金项目(11901421)；山西省自然科学基金面上项目(201901D111254)

杨丽丽(1997-)，女，山西临汾人，硕士生，主要从事量子信息与量子计算方面的研究(E-mail:1043177300@qq.com)；

*王银珠(1977-)，男，山西朔州人，副教授，博士，主要从事量子信息与量子计算方面的研究(E-mail:2006wang.yinzhu@163.com).