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一种与开放系统初态无关的非马尔科夫度量

贺志 蒋登魁 李艳

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一种与开放系统初态无关的非马尔科夫度量

贺志, 蒋登魁, 李艳

Non-Markovian measure independent of initial states of open systems

He Zhi, Jiang Deng-Kui, Li Yan
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  • 近年来, 量化开放量子系统中的非马尔科夫效应已经成为了量子消相干控制领域研究中的一个重要科学问题. 本文对于单个开放的两能级系统, 将基于量子Fisher信息的非马尔科夫度量从系统初态为纯态的情况推广到系统初态为任意态的情况. 作为该非马尔科夫度量的应用, 分别研究了利用量子Fisher信息在检测一个两能级系统受到零温度振幅耗散通道、相位衰减通道和随机幺正通道作用时对应非马尔可夫过程发生满足的条件. 研究结果显示: 一个相位参数的量子Fisher信息对这三种衰减通道的非马尔科夫过程发生所满足的条件与系统初态的选择是无关的. 进一步, 对于振幅耗散通道和相位衰减通道, 非马尔科夫过程发生的条件同基于迹距离、映射的可分性、量子互信息和量子Fisher信息矩阵等给出的条件是等价的. 如预期的一样, 对于振幅耗散通道情况且选择系统初态为最优化纯态时, 相应的结果正是Lu等获得的结果(Lu X M, Wang X G, Sun C P 2010 Phys. Rev. A 82 042103); 而对于随机幺正通道, 其马尔可夫过程发生的条件同基于迹距离、映射的可分性、量子互信息和量子Fisher信息矩阵等给出的条件是不完全等价的. 另外, 得到了一个有趣的关系: 在这三种耗散通道模型中系统演化态的量子$l_1$范数相干性的平方正好等于相位参数的量子Fisher信息. 总之, 本文得到的结果不仅完善了用量子Fisher信息来检测开放系统中非马尔科夫效应的应用范围, 同时也进一步彰显了量子Fisher信息在量子信息处理中独特的作用.
    In recent years, quantifying non-Markovian effect in open quantum system has become an important subject in the quantum decoherence control field. In this paper, a non-Markovian measure independent of the initial state of open system is proposed, thereby extending non-Markovian measure based on quantum Fisher information from the case where the initial state of the system is a pure state to the case where the initial state of the system is an arbitrary mixed state. As its application, the non-Markovian process is quantified by quantum Fisher information about a two-level system undergoing the three well-known dissipative channels, i.e. amplitude dissipative channel, phase damping channel, and random unitary channel. The results show that the conditions of non-Markovian processes in the three dissipative channels are independent of the selection of the initial state of the system by means of the quantum Fisher information of a phase parameter. Further, for amplitude dissipation channel and phase damping channel, the conditions for the non-Markovian processes to occur are equivalent to those given by trace distance, divisibility, quantum mutual information, quantum Fisher-information matrix, et al. As expected, for the case of amplitude dissipation channel, the corresponding results can reduce to the one in other paper (Lu X M, Wang X G, Sun C P 2010 Phys. Rev. A 82 042103) by selecting the initial state of the system as an optimal pure state. However, for random unitary channel, the conditions of non-Markovian process are not equivalent to those for other measures. In addition, we also obtain an interesting relationship between quantum Fisher information and quantum coherence of the open system in the three dissipative channels, namely the square of quantum $l_1$ coherence for the evolved state of system is exactly equal to the quantum Fisher information of the phase parameter. In a word, the obtained results not only improve the application scope of using the quantum Fisher information to detect non-Markovian effects in open systems, but also further highlight its important role in quantum information processing.
      通信作者: 贺志, hz9209@126.com
    • 基金项目: 中国博士后科学基金(批准号: 2017M622582)、湖南省教育厅重点项目(批准号: 19A339)、湖南省自然科学基金(批准号: 2020JJ4443, 2019JJ50400)、江西省教育厅科研基金(批准号: GJJ181086)和湖南省光电信息集成与光学制造技术重点实验室资助的课题
      Corresponding author: He Zhi, hz9209@126.com
    • Funds: Project supported by the China Postdoctoral Science Foundation (Grant No. 2017M622582), the Research Foundation of Education Bureau of Hunan Province, China (Grant No. 19A339), the Natural Science Foundation of Hunan Province, China (Grant Nos. 2020JJ4443, 2019JJ50400), the Scientific Research Fundation of the Education Department of Jiangxi Province, China (Grant No. GJJ181086), and the Key Laboratory of Photoelectric Information Integration and Optical Manufacturing Technology of Hunan Province, China
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    Breuer H P, Petruccione F 2002 The theory of Open Quantum Systems (Oxford: Oxford University Press) pp461–472

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    Buluta I, Ashhab S, Nori F 2011 Rep. Prog. Phys. 74 104401Google Scholar

    [3]

    Rivas A, Huelga S F, Plenio M B 2014 Rep. Prog. Phys. 77 094001Google Scholar

    [4]

    Breuer H P, Laine E M, Piilo J, Vacchini B 2016 Rev. Mod. Phys. 88 021002Google Scholar

    [5]

    Intravaia F, Behunin R O, Henkel C, Busch K, Dalvit D A R 2016 Phys. Rev. A 94 042114Google Scholar

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    Bellomo B, LoFranco R, Compagno G 2007 Phys. Rev. Lett. 99 160502Google Scholar

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    Zhang Y J, Man Z X, Xia Y J 2009 Eur. Phys. J. D 55 173Google Scholar

    [8]

    韩伟, 崔文凯, 张英杰, 夏云杰 2012 物理学报 61 230302Google Scholar

    Han W, Cui W K, Zhang Y J, Xia Y J 2012 Acta Phys. Sin. 61 230302Google Scholar

    [9]

    Xiao X, Fang M F, Hu Y M 2011 Phys. Scr. 84 045011Google Scholar

    [10]

    蔡诚俊, 方卯发, 肖兴, 黄江 2012 物理学报 61 210303Google Scholar

    Cai C J, Fang M F, Xiao X, Huang J 2012 Acta Phys. Sin. 61 210303Google Scholar

    [11]

    He Z, Huang B Y, Nie J J 2021 Laser Phys. Lett. 18 125202Google Scholar

    [12]

    Chin A W, Huelga S F, Plenio M B 2012 Phys. Rev. Lett. 109 233601Google Scholar

    [13]

    Berrada K 2013 Phys. Rev. A 88 035806Google Scholar

    [14]

    Breuer H P, Laine E M, Piilo J 2009 Phys. Rev. Lett. 103 210401Google Scholar

    [15]

    Xu Z Y, Yang W L, Feng M 2010 Phys. Rev. A 81 044105Google Scholar

    [16]

    Li J G, Zou J, Shao B 2010 Phys. Rev. A 81 062124Google Scholar

    [17]

    Rivas A, Huelga S F, Plenio M B 2010 Phys. Rev. Lett. 105 050403Google Scholar

    [18]

    Lu X M, Wang X G, Sun C P 2010 Phys. Rev. A 82 042103Google Scholar

    [19]

    Luo S, Fu S, Song H 2012 Phys. Rev. A 86 044101Google Scholar

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    Liu B H, Li L, Huang Y F, Li C F, Guo G C, Laine E M, Breuer H P, Piilo J 2011 Nat. Phys. 7 931Google Scholar

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    Tang J S, Li C F, Li Y L, Zou X B, Guo G C 2012 Europhys. Lett. 97 10002Google Scholar

    [22]

    Lorenzo S, Plastina F, Paternostro M 2013 Phys. Rev. A 88 020102Google Scholar

    [23]

    Liu J, Lu X M, Wang X G 2013 Phys. Rev. A 87 042103Google Scholar

    [24]

    Bylicka B, Chruscinski D, Maniscalco S 2014 Sci. Rep. 4 5720

    [25]

    Fanchini F F, Karpat G, Cakmak B, Castelano L K, et al. 2014 Phys. Rev. Lett. 112 210402Google Scholar

    [26]

    Song H, Luo S, Hong Y 2015 Phys. Rev. A 91 042110Google Scholar

    [27]

    Chen S L, Lambert N, Li C M, Miranowicz A, Chen Y N, Nori F 2016 Phys. Rev. Lett. 116 020503Google Scholar

    [28]

    贺志, 李莉, 姚春梅, 李艳 2015 物理学报 64 140302Google Scholar

    He Z, Li L, Yao C, Li Y 2015 Acta Phys. Sin. 64 140302Google Scholar

    [29]

    Luo Y, Li Y 2019 Chin. Phys. B 28 040301Google Scholar

    [30]

    Shao L H, Zhang Y R, Luo Y, Xi Z, Fei S M 2020 Laser Phys. Lett. 17 015202Google Scholar

    [31]

    Jahromi H R, Mahdavipour K, Shadfar M K, Lo Franco R 2020 Phys. Rev. A 102 022221Google Scholar

    [32]

    Sun L, Li J P, Tao Y H, Li L S 2022 Int. J. Theor. Phys. 61 134Google Scholar

    [33]

    Hou S C, Yi X X, Yu S X, Oh C H 2011 Phys. Rev. A 83 062115Google Scholar

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    Chruscinski D, Maniscalco 2014 Phys. Rev. Lett. 112 120404

    [35]

    Hall M J W, Cresser J D, Li L, Andersson E 2014 Phys. Rev. A 89 042120Google Scholar

    [36]

    He Z, Yao C, Zou J 2014 Phys. Rev. A 90 042101Google Scholar

    [37]

    Dhar H S, Bera M N, Adesso G 2015 Phys. Rev. A 91 032115Google Scholar

    [38]

    Paula F M, Obando P C, Sarandy M S 2016 Phys. Rev. A 93 042337Google Scholar

    [39]

    Haseli S, Karpat G, Salimi S 2014 Phys. Rev. A 90 052118Google Scholar

    [40]

    He Z, Zeng H S, Li Y, Wang Q, Yao C 2017 Phys. Rev. A 96 022106Google Scholar

    [41]

    Zeng H S, Tang N, Zheng Y P, Wang G Y 2011 Phys. Rev. A 84 032118Google Scholar

    [42]

    Braunstein S L, Caves C M 1994 Phys. Rev. Lett. 72 3439Google Scholar

    [43]

    钟伟 2014 博士学位论文 (杭州: 浙江大学)

    Zhong W 2014 Ph. D. Dissertation (Hangzhou: Zhejiang University) (in Chinese)

    [44]

    Liu J, Xiong H N, Song F, Wang X 2014 Physica A 410 167Google Scholar

    [45]

    Liu J, Jing X, Zhong W, Wang X 2014 Commun. Theor. Phys. 61 45Google Scholar

    [46]

    Zhong W, Sun Z, Ma J, Wang X, Nori F 2013 Phys. Rev. A 87 022337Google Scholar

    [47]

    Chruscinski D, Wudarski F 2013 Phys. Lett. A 377 1425Google Scholar

    [48]

    Jiang M, Luo S 2013 Phys. Rev. A 88 034101

    [49]

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

    [50]

    Zhao J L, Chen D X, Zhang Y, Fang Y L, Yang M, Wu Q C, Yang C P 2021 Phys. Rev. A 104 062608Google Scholar

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    陆晓铭 2011 博士学位论文 (杭州: 浙江大学)

    Lu X M 2011 Ph. D. Dissertation (Hangzhou: Zhejiang University) (in Chinese)

  • 图 1  在不同系统初态(这里用不同初态对应的非对角元值$\left| {\rho _{{\rm{eg}}} \left( 0 \right)} \right|$来表示)下$ \mathcal{N}_{{\rm{QFI}}} $随参数$ \lambda $的变化

    Fig. 1.  Non-Markovianity $ \mathcal{N}_{{\rm{QFI}}} $ as a function of $ \lambda $ for different initial states of system denoted by their off-diagonal elements $ \left| {\rho _{{\rm{eg}}} \left( 0 \right)} \right| $

    表 1  几种流行的非马尔科夫度量对不同$\alpha$值的马尔科夫条件比较

    Table 1.  Comparisons of the Markovian conditions for some popular non-Markovian measures under different $\alpha$

    不同的$\alpha$值 $\alpha=1/4$ $\alpha=1/3$ $\alpha=2/5$
    映射的可分性 $p_0 (t) \geqslant 1/3$ $p_0 (t) \geqslant 1/4$ $p_0 (t) \geqslant 0.583$
    迹距离 $p_0 (t) \geqslant 1/3$ $p_0 (t) \geqslant 1/4$ $p_0 (t) \geqslant 0.375$
    量子互信息 $p_0 (t) \geqslant 0.261$ $p_0 (t) \geqslant 1/4$ $p_0 (t) \geqslant 0.258$
    量子Fisher
    信息矩阵
    $p_0 (t) \geqslant 1/3$ $p_0 (t) \geqslant 1/4$ $p_0 (t) \geqslant 0.328$
    量子Fisher信息 $p_0 (t) \geqslant 1/3$ $p_0 (t) \geqslant 1/4$ $p_0 (t) \geqslant 0.167$
    下载: 导出CSV
  • [1]

    Breuer H P, Petruccione F 2002 The theory of Open Quantum Systems (Oxford: Oxford University Press) pp461–472

    [2]

    Buluta I, Ashhab S, Nori F 2011 Rep. Prog. Phys. 74 104401Google Scholar

    [3]

    Rivas A, Huelga S F, Plenio M B 2014 Rep. Prog. Phys. 77 094001Google Scholar

    [4]

    Breuer H P, Laine E M, Piilo J, Vacchini B 2016 Rev. Mod. Phys. 88 021002Google Scholar

    [5]

    Intravaia F, Behunin R O, Henkel C, Busch K, Dalvit D A R 2016 Phys. Rev. A 94 042114Google Scholar

    [6]

    Bellomo B, LoFranco R, Compagno G 2007 Phys. Rev. Lett. 99 160502Google Scholar

    [7]

    Zhang Y J, Man Z X, Xia Y J 2009 Eur. Phys. J. D 55 173Google Scholar

    [8]

    韩伟, 崔文凯, 张英杰, 夏云杰 2012 物理学报 61 230302Google Scholar

    Han W, Cui W K, Zhang Y J, Xia Y J 2012 Acta Phys. Sin. 61 230302Google Scholar

    [9]

    Xiao X, Fang M F, Hu Y M 2011 Phys. Scr. 84 045011Google Scholar

    [10]

    蔡诚俊, 方卯发, 肖兴, 黄江 2012 物理学报 61 210303Google Scholar

    Cai C J, Fang M F, Xiao X, Huang J 2012 Acta Phys. Sin. 61 210303Google Scholar

    [11]

    He Z, Huang B Y, Nie J J 2021 Laser Phys. Lett. 18 125202Google Scholar

    [12]

    Chin A W, Huelga S F, Plenio M B 2012 Phys. Rev. Lett. 109 233601Google Scholar

    [13]

    Berrada K 2013 Phys. Rev. A 88 035806Google Scholar

    [14]

    Breuer H P, Laine E M, Piilo J 2009 Phys. Rev. Lett. 103 210401Google Scholar

    [15]

    Xu Z Y, Yang W L, Feng M 2010 Phys. Rev. A 81 044105Google Scholar

    [16]

    Li J G, Zou J, Shao B 2010 Phys. Rev. A 81 062124Google Scholar

    [17]

    Rivas A, Huelga S F, Plenio M B 2010 Phys. Rev. Lett. 105 050403Google Scholar

    [18]

    Lu X M, Wang X G, Sun C P 2010 Phys. Rev. A 82 042103Google Scholar

    [19]

    Luo S, Fu S, Song H 2012 Phys. Rev. A 86 044101Google Scholar

    [20]

    Liu B H, Li L, Huang Y F, Li C F, Guo G C, Laine E M, Breuer H P, Piilo J 2011 Nat. Phys. 7 931Google Scholar

    [21]

    Tang J S, Li C F, Li Y L, Zou X B, Guo G C 2012 Europhys. Lett. 97 10002Google Scholar

    [22]

    Lorenzo S, Plastina F, Paternostro M 2013 Phys. Rev. A 88 020102Google Scholar

    [23]

    Liu J, Lu X M, Wang X G 2013 Phys. Rev. A 87 042103Google Scholar

    [24]

    Bylicka B, Chruscinski D, Maniscalco S 2014 Sci. Rep. 4 5720

    [25]

    Fanchini F F, Karpat G, Cakmak B, Castelano L K, et al. 2014 Phys. Rev. Lett. 112 210402Google Scholar

    [26]

    Song H, Luo S, Hong Y 2015 Phys. Rev. A 91 042110Google Scholar

    [27]

    Chen S L, Lambert N, Li C M, Miranowicz A, Chen Y N, Nori F 2016 Phys. Rev. Lett. 116 020503Google Scholar

    [28]

    贺志, 李莉, 姚春梅, 李艳 2015 物理学报 64 140302Google Scholar

    He Z, Li L, Yao C, Li Y 2015 Acta Phys. Sin. 64 140302Google Scholar

    [29]

    Luo Y, Li Y 2019 Chin. Phys. B 28 040301Google Scholar

    [30]

    Shao L H, Zhang Y R, Luo Y, Xi Z, Fei S M 2020 Laser Phys. Lett. 17 015202Google Scholar

    [31]

    Jahromi H R, Mahdavipour K, Shadfar M K, Lo Franco R 2020 Phys. Rev. A 102 022221Google Scholar

    [32]

    Sun L, Li J P, Tao Y H, Li L S 2022 Int. J. Theor. Phys. 61 134Google Scholar

    [33]

    Hou S C, Yi X X, Yu S X, Oh C H 2011 Phys. Rev. A 83 062115Google Scholar

    [34]

    Chruscinski D, Maniscalco 2014 Phys. Rev. Lett. 112 120404

    [35]

    Hall M J W, Cresser J D, Li L, Andersson E 2014 Phys. Rev. A 89 042120Google Scholar

    [36]

    He Z, Yao C, Zou J 2014 Phys. Rev. A 90 042101Google Scholar

    [37]

    Dhar H S, Bera M N, Adesso G 2015 Phys. Rev. A 91 032115Google Scholar

    [38]

    Paula F M, Obando P C, Sarandy M S 2016 Phys. Rev. A 93 042337Google Scholar

    [39]

    Haseli S, Karpat G, Salimi S 2014 Phys. Rev. A 90 052118Google Scholar

    [40]

    He Z, Zeng H S, Li Y, Wang Q, Yao C 2017 Phys. Rev. A 96 022106Google Scholar

    [41]

    Zeng H S, Tang N, Zheng Y P, Wang G Y 2011 Phys. Rev. A 84 032118Google Scholar

    [42]

    Braunstein S L, Caves C M 1994 Phys. Rev. Lett. 72 3439Google Scholar

    [43]

    钟伟 2014 博士学位论文 (杭州: 浙江大学)

    Zhong W 2014 Ph. D. Dissertation (Hangzhou: Zhejiang University) (in Chinese)

    [44]

    Liu J, Xiong H N, Song F, Wang X 2014 Physica A 410 167Google Scholar

    [45]

    Liu J, Jing X, Zhong W, Wang X 2014 Commun. Theor. Phys. 61 45Google Scholar

    [46]

    Zhong W, Sun Z, Ma J, Wang X, Nori F 2013 Phys. Rev. A 87 022337Google Scholar

    [47]

    Chruscinski D, Wudarski F 2013 Phys. Lett. A 377 1425Google Scholar

    [48]

    Jiang M, Luo S 2013 Phys. Rev. A 88 034101

    [49]

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

    [50]

    Zhao J L, Chen D X, Zhang Y, Fang Y L, Yang M, Wu Q C, Yang C P 2021 Phys. Rev. A 104 062608Google Scholar

    [51]

    陆晓铭 2011 博士学位论文 (杭州: 浙江大学)

    Lu X M 2011 Ph. D. Dissertation (Hangzhou: Zhejiang University) (in Chinese)

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出版历程
  • 收稿日期:  2022-05-26
  • 修回日期:  2022-08-09
  • 上网日期:  2022-10-22
  • 刊出日期:  2022-11-05

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