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相干与路径信息

吕鑫

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相干与路径信息

吕鑫

Coherence and path information

Lü Xin
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  • 近年来, 随着对相干性量化的发展, 相干与路径信息间的互补关系渐渐引起人们注意. 这样的互补关系不仅在基础量子力学方面有重要的理论意义, 同时也在量子信息技术中有实际应用. 本文从Bures距离和明确量子态区分出发, 系统地研究了二路径干涉仪中的相干与路径信息, 并建立了一个全新的互补关系. 与已知的类似工作相比, 本文的研究更为一般: 既考虑纯态, 也考虑混合态; 既探讨了干涉仪本身的路径不对称性所提供的路径可预测性, 也探讨了因路径探测器而生的路径可区分度. 具体地说, 路径可预测性的讨论只依赖于密度矩阵的半正定性质, 而路径可区分度的讨论还需要利用保真度和Schur-Weyl不等式等工具.
    Complementarity, or more specifically, the wave-particle duality, and quantum coherence are two fundamental concepts in quantum mechanics. Recently, motivated by the progress of the quantification of quantum coherence, the complementary relation between coherence and path information is investigated by many authors, and various duality relations between them are established. Such relations not only provide insights into fundamental problems of quantum mechanics, such as the understanding of quantum coherence and wave-particle duality; but also are important in applications of quantum technologies. In this paper, based on the Bures distance and unambiguous quantum state discrimination, systematic analysis of the complementarity between the quantum coherence and path information in two path interferometers is carried out. Similarly as other related works, the wave aspect, or the visibility of the interferometer, is quantified by the l1-norm measure of quantum coherence, and the path information is considered via unambiguous quantum state discrimination. In this way, a novel duality relation in two path interferometers is obtained. Compared with known results, our work considers mixed states as well as pure states; considers the path predictability resulting from the intrinsic path asymmetry of the quantum state, as well as the path distinguishability resulting from the use of a which-path detector. Therefore, our work systematically generalizes known results in two path interferometers by removing all the unnecessary restrictions. Specifically, the most general form of quantum states in two path interferometers is considered and the duality relation between quantum coherence and path information is proved based on the positivity property of density matrices. The cases of path predictability and path distinguishability are considered separately. For path predictability, the proof is straightforward; whereas some advanced mathematical techniques, such as the Schur-Weyl inequality, properties of the fidelity and properties of positive matrices, are required in order to give a rigorous proof of the duality relation between coherence and path distinguishability. Concrete examples are provided to illustrate the abstract method and results. Our work concerns about two path interferometers exclusively and depends heavily that the dimensionality is two, therefore it would be an interesting task to generalize the results in this paper to n-path interferometers.
      通信作者: 吕鑫, lyuxinlps@outlook.com
    • 基金项目: 贵州省教育厅青年科技人才成长项目(批准号: QJHKYZ[2018]377)、六盘水师范学院高层次人才科研启动基金(批准号: LPSSYKYJJ201813)、六盘水师范学院重点专业建设项目(批准号: LSZDZY2018-03)和贵州省科技厅科技平台及人才团队计划项目(批准号: QKH*Platform&Talent[2018]5777)资助的课题
      Corresponding author: Lü Xin, lyuxinlps@outlook.com
    • Funds: Project supported by the Young Scientific Talents Growth Project of the Department of Education of Guizhou Province (Grant No. QJHKYZ[2018]377), Research Fund for High-level Talents of Liupanshui Normal University (Grant No. LPSSYKYJJ201813), Key Disciplines Construction Project of Liupanshui Normal University (Grant No. LSZDZY2018-03) and panshui Normal University (LPSSYKYJJ201813), Science and Technology Platform and Talent team Project of Science and Technology Department of Guizhou Province (Grant No. QKH*Platform&Talent[2018]5777)
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    Greenberger D M, Yasin A 1988 Phys. Lett. A 128 391Google Scholar

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    Jaeger G, Shimony A, Vaidman L 1995 Phys. Rev. A 51 54Google Scholar

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    Englert B G, Bergou J A 2000 Opt. Commun. 179 337Google Scholar

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    Dürr S 2001 Phys. Rev. A 64 42113Google Scholar

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    Englert B G, Kaszlikowski D, Kwek L C, Chee W H 2008 Int. J. Quantum Inf. 06 129Google Scholar

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    Li L, Liu N L, Yu S 2012 Phys. Rev. A 85 054101Google Scholar

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    Bera M N, Qureshi T, Siddiqui M A, Pati A k 2015 Phys. Rev. A 92 012118Google Scholar

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    Bagan E, Bergou J A, Cottrell S S, Hillery M 2016 Phys. Rev. Lett. 116 160406Google Scholar

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    Bagan E, Calsamiglia J, Bergou J A, Hillery M 2018 Phys. Rev. Lett. 120 050402Google Scholar

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    傅双双, 骆顺龙, 孙源 2019 物理学报 68 030301Google Scholar

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    Mei M, Weitz M 2001 Phys. Rev. Lett. 86 559Google Scholar

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    Trifonov A, Björk G, Söderholm, Tsegaye T 2002 Eur. Phys. J. D 18 251

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    Tang J S, Li Y L, Xu X Y, Xiang G Y, Li C F, Guo G C 2012 Nat. Photonics 6 600Google Scholar

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    Peruzzo A, Shadbolt P, Brunner N, Popescu S, O’Brien J L 2012 Science 338 634Google Scholar

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    Girolami D 2014 Phys. Rev. Lett. 113 170401Google Scholar

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    Streltsov A, Singh U, Dhar H S, Bera M N, Adesso G 2015 Phys. Rev. Lett. 115 020403Google Scholar

    [34]

    Pires D P, Celeri L C, Soares-Pinto D O 2015 Phys. Rev. A 91 042330Google Scholar

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    Yao Y, Xiao X, Ge L, Sun C P 2015 Phys. Rev. A 92 022112Google Scholar

    [36]

    Winter A, Yang D 2016 Phys. Rev. Lett. 116 120404Google Scholar

    [37]

    Ma J, Yadin B, Girolami D, Vedral V, Gu M 2016 Phys. Rev. Lett. 116 160407Google Scholar

    [38]

    Chang L, Luo S, Sun Y 2017 Commun. Theor. Phys. 68 565Google Scholar

    [39]

    Streltsov A, Adesso G, Plenio M B 2017 Rev. Mod. Phys. 89 041003Google Scholar

    [40]

    Luo S, Sun Y 2017 Phys. Rev. A 96 022130Google Scholar

    [41]

    Luo S, Sun Y 2017 Phys. Rev. A 96 022136Google Scholar

    [42]

    Yao Y, Dong G H, Xiao X, Li M, Sun C P 2017 Phys. Rev. A 96 052322Google Scholar

    [43]

    Jin Z X, Fei S M 2018 Phys. Rev. A 97 062342Google Scholar

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    Bures D 1969 Trans. Amer. Math. Soc. 135 199

    [45]

    Bengtsson I, Życzkowski K 2006 Geometry of Quantum States: An Introduction to Quantum Entanglement (Cambridge: Cambridge University Press) p258, pp396–401

    [46]

    Nielsen, M A, Chuang I L 2000 Quantum Computation and Quantum Information (10th Anniversary Edition) (New York: Cambridge University Press) pp60–111, 399–416

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    Chefles A 2000 Contemp. Phys. 41 401Google Scholar

    [48]

    Bergou J A, Herzog U, Hillery M 2004 Discrimination of Quantum States in Quantum State Estimation (Berlin/ Heidelberg: Springer) pp417–465

    [49]

    Bae J, Kwek L C 2015 J. Phys. A: Math. Gen. 48 083001Google Scholar

    [50]

    Feng Y, Duan R, Ying M 2004 Phys. Rev. A 70 012308Google Scholar

    [51]

    Uhlmann A 2000 Phys. Rev. A 62 032307Google Scholar

    [52]

    Jozsa R 1994 J. Mod. Opt. 41 2315Google Scholar

    [53]

    Bhatia R 2007 Positive Definite Matrices (Princeton/Oxford: Princeton University Press) pp5–12

    [54]

    Bellman R 1968 Linear Algebra Appl. 1 321Google Scholar

    [55]

    König H 1979 Studia Math. 65 141Google Scholar

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    König H 1986 Eigenvalue Distribution of Compact Operators (Basel: Birkhauser) p35

    [57]

    Pietsch A 1987 Eigenvalues and s-numbers (Leipzig: Geest & Portig) p155

    [58]

    Reed M, Simon B 1978 Methods of Modern Mathematical Physics IV: Analysis of Operators (New York: Academic) p318

  • 图 1  明确量子态区分在有噪信道中的体现

    Fig. 1.  Unambiguous quantum state discrimination in a noisy quantum channel

    图 2  利用路径探测器(WPD)的Mach-Zehnder干涉仪

    Fig. 2.  The Mach-Zehnder interferometer with a which-path detector

  • [1]

    Bohr N 1928 Nature 121 580Google Scholar

    [2]

    Scully M O, Englert B G, Walther H 1991 Nature 351 111Google Scholar

    [3]

    Wootters W K, Zurek W H 1979 Phys. Rev. D 19 473Google Scholar

    [4]

    Greenberger D M, Yasin A 1988 Phys. Lett. A 128 391Google Scholar

    [5]

    Jaeger G, Shimony A, Vaidman L 1995 Phys. Rev. A 51 54Google Scholar

    [6]

    Englert B G 1996 Phys. Rev. Lett. 77 2154Google Scholar

    [7]

    Englert B G, Bergou J A 2000 Opt. Commun. 179 337Google Scholar

    [8]

    Dürr S 2001 Phys. Rev. A 64 42113Google Scholar

    [9]

    Englert B G, Kaszlikowski D, Kwek L C, Chee W H 2008 Int. J. Quantum Inf. 06 129Google Scholar

    [10]

    Li L, Liu N L, Yu S 2012 Phys. Rev. A 85 054101Google Scholar

    [11]

    Huang J H, Wölk S, Zhu S Y, Zubairy M S 2013 Phys. Rev. A 87 022107Google Scholar

    [12]

    Jia A A, Huang J H, Feng W, Zhang T C, Zhu S Y 2014 Chin. Phys. B 23 030307Google Scholar

    [13]

    Angelo R M, Ribeiro A D 2015 Found. Phys. 45 1407Google Scholar

    [14]

    Bera M N, Qureshi T, Siddiqui M A, Pati A k 2015 Phys. Rev. A 92 012118Google Scholar

    [15]

    Bagan E, Bergou J A, Cottrell S S, Hillery M 2016 Phys. Rev. Lett. 116 160406Google Scholar

    [16]

    Bagan E, Calsamiglia J, Bergou J A, Hillery M 2018 Phys. Rev. Lett. 120 050402Google Scholar

    [17]

    傅双双, 骆顺龙, 孙源 2019 物理学报 68 030301Google Scholar

    Fu S, Luo S, Sun Y 2019 Acta Phys. Sin. 68 030301Google Scholar

    [18]

    Summhammer J, Rauch H, Tuppinger D 1987 Phys. Rev. A 36 4447Google Scholar

    [19]

    Dürr S, Nonn T, Rempe G 1998 Nature 395 33Google Scholar

    [20]

    Dürr S, Nonn T, Rempe G 1998 Phys. Rev. Lett. 81 5705Google Scholar

    [21]

    Mei M, Weitz M 2001 Phys. Rev. Lett. 86 559Google Scholar

    [22]

    Trifonov A, Björk G, Söderholm, Tsegaye T 2002 Eur. Phys. J. D 18 251

    [23]

    Tang J S, Li Y L, Xu X Y, Xiang G Y, Li C F, Guo G C 2012 Nat. Photonics 6 600Google Scholar

    [24]

    Peruzzo A, Shadbolt P, Brunner N, Popescu S, O’Brien J L 2012 Science 338 634Google Scholar

    [25]

    Kaiser F, Coudreau T, Milman P, Ostrowsky D B, Tanzilli S 2012 Science 338 637Google Scholar

    [26]

    Tang J S, Li Y L, Li C F, Guo G C 2013 Phys. Rev. A 88 014103Google Scholar

    [27]

    Huang J H, Liu H Y, Gao J R, Zubairy M S, Zhu S Y 2013 Phys. Rev. A 88 013828Google Scholar

    [28]

    Jia A A, Huang J H, Zhang T C, Zhu S Y 2014 Phys. Rev. A 89 042103Google Scholar

    [29]

    Wang Z, Tian Y, Yang C, Zhang P, Li G, Zhang T 2016 Phys. Rev. A 94 062124Google Scholar

    [30]

    Levi F, Mintert F 2014 New J. Phys. 16 033007Google Scholar

    [31]

    Baumgratz T, Cramer M, Plenio M B 2014 Phys. Rev. Lett. 113 140401Google Scholar

    [32]

    Girolami D 2014 Phys. Rev. Lett. 113 170401Google Scholar

    [33]

    Streltsov A, Singh U, Dhar H S, Bera M N, Adesso G 2015 Phys. Rev. Lett. 115 020403Google Scholar

    [34]

    Pires D P, Celeri L C, Soares-Pinto D O 2015 Phys. Rev. A 91 042330Google Scholar

    [35]

    Yao Y, Xiao X, Ge L, Sun C P 2015 Phys. Rev. A 92 022112Google Scholar

    [36]

    Winter A, Yang D 2016 Phys. Rev. Lett. 116 120404Google Scholar

    [37]

    Ma J, Yadin B, Girolami D, Vedral V, Gu M 2016 Phys. Rev. Lett. 116 160407Google Scholar

    [38]

    Chang L, Luo S, Sun Y 2017 Commun. Theor. Phys. 68 565Google Scholar

    [39]

    Streltsov A, Adesso G, Plenio M B 2017 Rev. Mod. Phys. 89 041003Google Scholar

    [40]

    Luo S, Sun Y 2017 Phys. Rev. A 96 022130Google Scholar

    [41]

    Luo S, Sun Y 2017 Phys. Rev. A 96 022136Google Scholar

    [42]

    Yao Y, Dong G H, Xiao X, Li M, Sun C P 2017 Phys. Rev. A 96 052322Google Scholar

    [43]

    Jin Z X, Fei S M 2018 Phys. Rev. A 97 062342Google Scholar

    [44]

    Bures D 1969 Trans. Amer. Math. Soc. 135 199

    [45]

    Bengtsson I, Życzkowski K 2006 Geometry of Quantum States: An Introduction to Quantum Entanglement (Cambridge: Cambridge University Press) p258, pp396–401

    [46]

    Nielsen, M A, Chuang I L 2000 Quantum Computation and Quantum Information (10th Anniversary Edition) (New York: Cambridge University Press) pp60–111, 399–416

    [47]

    Chefles A 2000 Contemp. Phys. 41 401Google Scholar

    [48]

    Bergou J A, Herzog U, Hillery M 2004 Discrimination of Quantum States in Quantum State Estimation (Berlin/ Heidelberg: Springer) pp417–465

    [49]

    Bae J, Kwek L C 2015 J. Phys. A: Math. Gen. 48 083001Google Scholar

    [50]

    Feng Y, Duan R, Ying M 2004 Phys. Rev. A 70 012308Google Scholar

    [51]

    Uhlmann A 2000 Phys. Rev. A 62 032307Google Scholar

    [52]

    Jozsa R 1994 J. Mod. Opt. 41 2315Google Scholar

    [53]

    Bhatia R 2007 Positive Definite Matrices (Princeton/Oxford: Princeton University Press) pp5–12

    [54]

    Bellman R 1968 Linear Algebra Appl. 1 321Google Scholar

    [55]

    König H 1979 Studia Math. 65 141Google Scholar

    [56]

    König H 1986 Eigenvalue Distribution of Compact Operators (Basel: Birkhauser) p35

    [57]

    Pietsch A 1987 Eigenvalues and s-numbers (Leipzig: Geest & Portig) p155

    [58]

    Reed M, Simon B 1978 Methods of Modern Mathematical Physics IV: Analysis of Operators (New York: Academic) p318

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出版历程
  • 收稿日期:  2019-11-03
  • 修回日期:  2020-01-21
  • 刊出日期:  2020-04-05

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