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