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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.
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Keywords:
- coherence /
- complementarity /
- wave-particle duality /
- unambiguous quantum state discrimination
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图 1 明确量子态区分在有噪信道中的体现
Figure 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
Figure 2. The Mach-Zehnder interferometer with a which-path detector
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[1] Bohr N 1928 Nature 121 580
Google Scholar
[2] Scully M O, Englert B G, Walther H 1991 Nature 351 111
Google Scholar
[3] Wootters W K, Zurek W H 1979 Phys. Rev. D 19 473
Google Scholar
[4] Greenberger D M, Yasin A 1988 Phys. Lett. A 128 391
Google Scholar
[5] Jaeger G, Shimony A, Vaidman L 1995 Phys. Rev. A 51 54
Google Scholar
[6] Englert B G 1996 Phys. Rev. Lett. 77 2154
Google Scholar
[7] Englert B G, Bergou J A 2000 Opt. Commun. 179 337
Google Scholar
[8] Dürr S 2001 Phys. Rev. A 64 42113
Google Scholar
[9] Englert B G, Kaszlikowski D, Kwek L C, Chee W H 2008 Int. J. Quantum Inf. 06 129
Google Scholar
[10] Li L, Liu N L, Yu S 2012 Phys. Rev. A 85 054101
Google Scholar
[11] Huang J H, Wölk S, Zhu S Y, Zubairy M S 2013 Phys. Rev. A 87 022107
Google Scholar
[12] Jia A A, Huang J H, Feng W, Zhang T C, Zhu S Y 2014 Chin. Phys. B 23 030307
Google Scholar
[13] Angelo R M, Ribeiro A D 2015 Found. Phys. 45 1407
Google Scholar
[14] Bera M N, Qureshi T, Siddiqui M A, Pati A k 2015 Phys. Rev. A 92 012118
Google Scholar
[15] Bagan E, Bergou J A, Cottrell S S, Hillery M 2016 Phys. Rev. Lett. 116 160406
Google Scholar
[16] Bagan E, Calsamiglia J, Bergou J A, Hillery M 2018 Phys. Rev. Lett. 120 050402
Google Scholar
[17] 傅双双, 骆顺龙, 孙源 2019 68 030301
Google Scholar
Fu S, Luo S, Sun Y 2019 Acta Phys. Sin. 68 030301
Google Scholar
[18] Summhammer J, Rauch H, Tuppinger D 1987 Phys. Rev. A 36 4447
Google Scholar
[19] Dürr S, Nonn T, Rempe G 1998 Nature 395 33
Google Scholar
[20] Dürr S, Nonn T, Rempe G 1998 Phys. Rev. Lett. 81 5705
Google Scholar
[21] Mei M, Weitz M 2001 Phys. Rev. Lett. 86 559
Google 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 6 600
Google Scholar
[24] Peruzzo A, Shadbolt P, Brunner N, Popescu S, O’Brien J L 2012 Science 338 634
Google Scholar
[25] Kaiser F, Coudreau T, Milman P, Ostrowsky D B, Tanzilli S 2012 Science 338 637
Google Scholar
[26] Tang J S, Li Y L, Li C F, Guo G C 2013 Phys. Rev. A 88 014103
Google Scholar
[27] Huang J H, Liu H Y, Gao J R, Zubairy M S, Zhu S Y 2013 Phys. Rev. A 88 013828
Google Scholar
[28] Jia A A, Huang J H, Zhang T C, Zhu S Y 2014 Phys. Rev. A 89 042103
Google Scholar
[29] Wang Z, Tian Y, Yang C, Zhang P, Li G, Zhang T 2016 Phys. Rev. A 94 062124
Google Scholar
[30] Levi F, Mintert F 2014 New J. Phys. 16 033007
Google Scholar
[31] Baumgratz T, Cramer M, Plenio M B 2014 Phys. Rev. Lett. 113 140401
Google Scholar
[32] Girolami D 2014 Phys. Rev. Lett. 113 170401
Google Scholar
[33] Streltsov A, Singh U, Dhar H S, Bera M N, Adesso G 2015 Phys. Rev. Lett. 115 020403
Google Scholar
[34] Pires D P, Celeri L C, Soares-Pinto D O 2015 Phys. Rev. A 91 042330
Google Scholar
[35] Yao Y, Xiao X, Ge L, Sun C P 2015 Phys. Rev. A 92 022112
Google Scholar
[36] Winter A, Yang D 2016 Phys. Rev. Lett. 116 120404
Google Scholar
[37] Ma J, Yadin B, Girolami D, Vedral V, Gu M 2016 Phys. Rev. Lett. 116 160407
Google Scholar
[38] Chang L, Luo S, Sun Y 2017 Commun. Theor. Phys. 68 565
Google Scholar
[39] Streltsov A, Adesso G, Plenio M B 2017 Rev. Mod. Phys. 89 041003
Google Scholar
[40] Luo S, Sun Y 2017 Phys. Rev. A 96 022130
Google Scholar
[41] Luo S, Sun Y 2017 Phys. Rev. A 96 022136
Google Scholar
[42] Yao Y, Dong G H, Xiao X, Li M, Sun C P 2017 Phys. Rev. A 96 052322
Google Scholar
[43] Jin Z X, Fei S M 2018 Phys. Rev. A 97 062342
Google 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. 41 401
Google 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 083001
Google Scholar
[50] Feng Y, Duan R, Ying M 2004 Phys. Rev. A 70 012308
Google Scholar
[51] Uhlmann A 2000 Phys. Rev. A 62 032307
Google Scholar
[52] Jozsa R 1994 J. Mod. Opt. 41 2315
Google Scholar
[53] Bhatia R 2007 Positive Definite Matrices (Princeton/Oxford: Princeton University Press) pp5–12
[54] Bellman R 1968 Linear Algebra Appl. 1 321
Google Scholar
[55] König H 1979 Studia Math. 65 141
Google 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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