相干与信息守恒及其在Mach-Zehnder干涉中的应用

傅双双; 骆顺龙; 孙源

doi:10.7498/aps.68.20181778

摘要

自量子力学诞生以来, 相干性和互补性一直是被广泛而深入研究的两个重要课题. 随着量子信息近年来的发展, 人们引入了若干度量来定量地刻画相干性和互补性. 本文建立两个信息守恒关系式, 分别基于“Bures距离-保真度”和“对称-非对称”, 并且利用它们来刻画相干性和互补性. 具体来说, 首先从信息守恒的观点解释Bures距离和保真度的互补关系, 并由此自然推导出Mach-Zehnder 干涉仪中的Englert“干涉-路径”互补关系. 其次在量子态和信道相互作用的一般框架中讨论“对称-非对称”信息守恒关系, 并揭示其与Bohr互补性和量子相干性的内在联系. 最后, 在Mach-Zehnder干涉仪中探讨相干、退相干及互补性, 刻画两个信息守恒关系之间的密切联系.

关键词:

Abstract

Coherence and complementarity are two important themes in quantum mechanics, which have been widely and thoroughly investigated. Recently, with the rapid development of quantum information theory, various measures have been introduced for quantitatively studying the coherence and complementarity. However, most of these studies are independent of each other in that they focus on only one theme, for example, the wave-particle duality and Heisenberg uncertainty principle are usually regarded as manifestation of Bohr’s complementary principle, while coherence is a quantum feature closely related to quantum superposition. During the past few years, there has been a flurry of research interest in the study of quantum coherence from the quantum resource-theoretic point of view. In this paper, we establish two information conservation relations and employ them to characterize complementarity and quantum coherence. As an illustration of the main results, we discuss these two themes in the Mach-Zehnder interferometer. Our study reveals that these two quantum themes are closely related to each other. Our main results are listed as follows. Firstly, we establish two information conservation relations, one is based on " Bures distance versus fidelity” and the other based on " symmetry versus asymmetry”. Then we employ these information conservation relations to investigate coherence and complementarity. Specifically, we provide an explanation of the " Bures distance versus fidelity” trade-off relation from the information conservation perspective, establish the link between the information conservation relation and wave-particle duality, and derive the famous Englert inequality concerning " fringe visibility versus path distinguishability” from the information conservation relation. Furthermore, in the general framework of state-channel interaction, we derive " symmetry versus asymmetry” trade-off relation and explain it as an information conservation relation, reveal its intrinsic relations with coherence and complementarity. Lastly, we demonstrate that the two information conservation relations are closely interrelated, and we also discuss the coherence, decoherence and complementarity in the Mach-Zehnder interferometer, explicitly, we reveal that the Bures distance can be regarded as a lower bound of the asymmetry of state-channel interaction while fidelity is an upper bound of the symmetry of state-channel interaction. We expect that our information conservation relation can provide a unified framework for the study of coherence and complementarity.

Keywords:

作者及机构信息

1.
北京科技大学数理学院, 北京　100083

2.
中国科学院数学与系统科学研究院, 北京　100190

3.
中国科学院大学数学学院, 北京　100049

通信作者: 骆顺龙, luosl@amt.ac.cn

基金项目: 北京市自然科学基金(批准号: 1174017)、国家自然科学基金青年科学基金(批准号: 11605006)、国家自然科学基金(批准号: 11875317)、国家数学与交叉科学中心(批准号: Y029152K51)和中国科学院随机复杂结构与数据科学重点实验室(批准号: 2008DP173182)资助的课题.

Authors and contacts

1.
School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China

2.
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

3.
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

Corresponding author: Luo Shun-Long, luosl@amt.ac.cn

Funds: Project supported by the Natural Science Foundation of Beijing, China (Grant No. 1174017), the Young Scientists Fund of the National Natural Science Foundation of China (Grant No. 11605006), the National Natural Science Foundation of China (Grant No. 11875317), the National Center for Mathematics and Interdisciplinary Sciences, Chinese Academy of Sciences, China (Grant No. Y029152K51), and the Key Laboratory of Random Complex Structures and Data Science, Chinese Academy of Sciences, China (Grant No. 2008DP173182).

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参考文献

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施引文献

图 1 对称MZI (BS, 50 : 50分束器; PS, 相移器; WWD, 路径探测器)

Fig. 1. A schematic sketch of the symmetric Mach-Zehnder interferometer. BS, 50:50 beam splitter; PS, phase shifter; WWD, which-way detector.

下载: 全尺寸图片幻灯片

[1]	Bohr N 1937 Phil. Sci. 4 289 Google Scholar
[2]	Heisenberg W 1927 Zeit. Physik 43 172 Google Scholar
[3]	Heisenberg W 1930 The Physical Principles of the Quantum Theory (Chicago: The University of Chicago Press) pp13–39
[4]	Wootters W K, Zurek W H 1979 Phys. Rev. D 19 473 Google Scholar
[5]	Scully M O, Englert B G, Walther H 1991 Nature 351 111 Google Scholar
[6]	Mandel L 1991 Opt. Lett. 16 1882 Google Scholar
[7]	Jaeger G, Horne M A, Shimony A 1993 Phys. Rev. A 48 1023 Google Scholar
[8]	Englert B G 1996 Phys. Rev. Lett. 77 2154 Google Scholar
[9]	Busch P, Shilladay C 2006 Phys. Rep. 435 1 Google Scholar
[10]	Coles P J, Kaniewski J, Wehner S 2014 Nat. Commun. 5 5814 Google Scholar
[11]	Coles P J, Berta M, Tomamichel M, Wehner S 2017 Rev. Mod. Phys. 89 015002 Google Scholar
[12]	Jaeger G, Shimony A, Vaidmann L 1995 Phys. Rev. A 51 54 Google Scholar
[13]	Åberg J 2006 arXiv:quant-ph/0612146
[14]	Levi F, Mintert F 2014 New J. Phys. 16 033007 Google Scholar
[15]	Baumgratz T, Cramer M, Plenio M B 2014 Phys. Rev. Lett. 113 140401 Google Scholar
[16]	Girolami D 2014 Phys. Rev. Lett. 113 170401 Google Scholar
[17]	Streltsov A, Singh U, Dhar H S, Bera M N, Adesso G 2015 Phys. Rev. Lett. 115 020403 Google Scholar
[18]	Pires D P, Celeri L C, Soares-Pinto D O 2015 Phys. Rev. A 91 042330 Google Scholar
[19]	Yao Y, Xiao X, Ge L, Sun C P 2015 Phys. Rev. A 92 022112 Google Scholar
[20]	Winter A, Yang D 2016 Phys. Rev. Lett. 116 120404 Google Scholar
[21]	Ma J, Yadin B, Girolami D, Vedral V, Gu M 2016 Phys. Rev. Lett. 116 160407 Google Scholar
[22]	Chang L, Luo S, Sun Y 2017 Commun. Theor. Phys. 68 565 Google Scholar
[23]	Streltsov A, Adesso G, Plenio M B 2017 Rev. Mod. Phys. 89 041003 Google Scholar
[24]	Luo S, Sun Y 2017 Phys. Rev. A 96 022130 Google Scholar
[25]	Luo S, Sun Y 2017 Phys. Rev. A 96 022136 Google Scholar
[26]	Yao Y, Dong G H, Xiao X, Li M, Sun C P 2017 Phys. Rev. A 96 052322 Google Scholar
[27]	Zhao H, Yu C 2018 Sci. Rep. 8 299 Google Scholar
[28]	Jin Z X, Fei S M 2018 Phys. Rev. A 97 062342 Google Scholar
[29]	Horodecki M, Oppenheim J 2013 Nat. Commun. 4 2059 Google Scholar
[30]	Narasimhachar V, Gour G 2015 Nat. Commun. 6 7689 Google Scholar
[31]	Lloyd S 2011 J. Phys. Conf. Ser. 302 012037 Google Scholar
[32]	Marvian I, Spekkens R W 2014 Nat. Commun. 5 3821 Google Scholar
[33]	Marvian I, Spekkens R W 2016 Phys. Rev. A 94 052324 Google Scholar
[34]	Fang Y N, Dong G H, Zhou D L, Sun C P 2016 Commun. Theor. Phys. 65 423 Google Scholar
[35]	Yao Y, Dong G H., Xiao X, Sun C P 2016 Sci. Rep. 6 32010 Google Scholar
[36]	Bagan E, Bergou J A, Cottrell S S, Hillery M 2016 Phys. Rev. Lett. 116 160406 Google Scholar
[37]	Bera M N, Qureshi T, Siddiqui M A, Pati A K 2015 Phys. Rev. A 92 012118 Google Scholar
[38]	Hu M L, Hu X , Wang J C, Peng Yi, Zhang Y R, Fan H 2018 arXiv:1703.01852 [quant-ph]
[39]	Luo S, Sun Y 2018 Phys. Rev. A 98 012113 Google Scholar
[40]	Nielsen, M A, Chuang I L 2000 Quantum Computation and Quantum Information (10th Anniversary Edition) (New York: Cambridge University Press) pp60–111, 399–416
[41]	Bures D 1969 Trans. Amer. Math. Soc. 135 199-212 Google Scholar
[42]	Hubner M 1993 Phys. Lett. A 179 226 Google Scholar
[43]	Fuchs C A, Caves C M 1995 Open Sys. Inf. Dyn. 3 345 Google Scholar
[44]	Barnum H, Caves C M, Fuchs C A, Jozsa R, Schumacher B 1996 Phys. Rev. Lett. 76 2818 Google Scholar
[45]	Uhlmann A 2000 Phys. Rev. A 62 032307 Google Scholar
[46]	Dodd J L, Nielsen M A 2002 Phys. Rev. A 66 044301 Google Scholar
[47]	Luo S, Zhang Q 2004 Phys. Rev. A 69 032106 Google Scholar

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相干与信息守恒及其在Mach-Zehnder干涉中的应用