Non-Markovian measure independent of initial states of open systems

He Zhi; Jiang Deng-Kui; Li Yan

doi:10.7498/aps.71.20221053

Abstract

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.

Keywords:

Authors and contacts

1.
College of Mathematics and Physics, Hunan University of Arts and Science, Changde 415000, China
2.
College of Physics and Telecommunication, Nanchang Normal University, Nanchang 330032, China

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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References

[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 104401 Google Scholar
[3]	Rivas A, Huelga S F, Plenio M B 2014 Rep. Prog. Phys. 77 094001 Google Scholar
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[5]	Intravaia F, Behunin R O, Henkel C, Busch K, Dalvit D A R 2016 Phys. Rev. A 94 042114 Google Scholar
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[48]	Jiang M, Luo S 2013 Phys. Rev. A 88 034101
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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

Figure 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| $

DownLoad: Full-Size Img PowerPoint

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	$p_0 (t) \geqslant 1/3$	$p_0 (t) \geqslant 1/4$	$p_0 (t) \geqslant 0.167$

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[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 104401 Google Scholar
[3]	Rivas A, Huelga S F, Plenio M B 2014 Rep. Prog. Phys. 77 094001 Google Scholar
[4]	Breuer H P, Laine E M, Piilo J, Vacchini B 2016 Rev. Mod. Phys. 88 021002 Google Scholar
[5]	Intravaia F, Behunin R O, Henkel C, Busch K, Dalvit D A R 2016 Phys. Rev. A 94 042114 Google Scholar
[6]	Bellomo B, LoFranco R, Compagno G 2007 Phys. Rev. Lett. 99 160502 Google Scholar
[7]	Zhang Y J, Man Z X, Xia Y J 2009 Eur. Phys. J. D 55 173 Google Scholar
[8]	韩伟, 崔文凯, 张英杰, 夏云杰 2012 61 230302 Google Scholar Han W, Cui W K, Zhang Y J, Xia Y J 2012 Acta Phys. Sin. 61 230302 Google Scholar
[9]	Xiao X, Fang M F, Hu Y M 2011 Phys. Scr. 84 045011 Google Scholar
[10]	蔡诚俊, 方卯发, 肖兴, 黄江 2012 61 210303 Google Scholar Cai C J, Fang M F, Xiao X, Huang J 2012 Acta Phys. Sin. 61 210303 Google Scholar
[11]	He Z, Huang B Y, Nie J J 2021 Laser Phys. Lett. 18 125202 Google Scholar
[12]	Chin A W, Huelga S F, Plenio M B 2012 Phys. Rev. Lett. 109 233601 Google Scholar
[13]	Berrada K 2013 Phys. Rev. A 88 035806 Google Scholar
[14]	Breuer H P, Laine E M, Piilo J 2009 Phys. Rev. Lett. 103 210401 Google Scholar
[15]	Xu Z Y, Yang W L, Feng M 2010 Phys. Rev. A 81 044105 Google Scholar
[16]	Li J G, Zou J, Shao B 2010 Phys. Rev. A 81 062124 Google Scholar
[17]	Rivas A, Huelga S F, Plenio M B 2010 Phys. Rev. Lett. 105 050403 Google Scholar
[18]	Lu X M, Wang X G, Sun C P 2010 Phys. Rev. A 82 042103 Google Scholar
[19]	Luo S, Fu S, Song H 2012 Phys. Rev. A 86 044101 Google 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. 7 931 Google Scholar
[21]	Tang J S, Li C F, Li Y L, Zou X B, Guo G C 2012 Europhys. Lett. 97 10002 Google Scholar
[22]	Lorenzo S, Plastina F, Paternostro M 2013 Phys. Rev. A 88 020102 Google Scholar
[23]	Liu J, Lu X M, Wang X G 2013 Phys. Rev. A 87 042103 Google 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 210402 Google Scholar
[26]	Song H, Luo S, Hong Y 2015 Phys. Rev. A 91 042110 Google Scholar
[27]	Chen S L, Lambert N, Li C M, Miranowicz A, Chen Y N, Nori F 2016 Phys. Rev. Lett. 116 020503 Google Scholar
[28]	贺志, 李莉, 姚春梅, 李艳 2015 64 140302 Google Scholar He Z, Li L, Yao C, Li Y 2015 Acta Phys. Sin. 64 140302 Google Scholar
[29]	Luo Y, Li Y 2019 Chin. Phys. B 28 040301 Google Scholar
[30]	Shao L H, Zhang Y R, Luo Y, Xi Z, Fei S M 2020 Laser Phys. Lett. 17 015202 Google Scholar
[31]	Jahromi H R, Mahdavipour K, Shadfar M K, Lo Franco R 2020 Phys. Rev. A 102 022221 Google Scholar
[32]	Sun L, Li J P, Tao Y H, Li L S 2022 Int. J. Theor. Phys. 61 134 Google Scholar
[33]	Hou S C, Yi X X, Yu S X, Oh C H 2011 Phys. Rev. A 83 062115 Google 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 042120 Google Scholar
[36]	He Z, Yao C, Zou J 2014 Phys. Rev. A 90 042101 Google Scholar
[37]	Dhar H S, Bera M N, Adesso G 2015 Phys. Rev. A 91 032115 Google Scholar
[38]	Paula F M, Obando P C, Sarandy M S 2016 Phys. Rev. A 93 042337 Google Scholar
[39]	Haseli S, Karpat G, Salimi S 2014 Phys. Rev. A 90 052118 Google Scholar
[40]	He Z, Zeng H S, Li Y, Wang Q, Yao C 2017 Phys. Rev. A 96 022106 Google Scholar
[41]	Zeng H S, Tang N, Zheng Y P, Wang G Y 2011 Phys. Rev. A 84 032118 Google Scholar
[42]	Braunstein S L, Caves C M 1994 Phys. Rev. Lett. 72 3439 Google 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 410 167 Google Scholar
[45]	Liu J, Jing X, Zhong W, Wang X 2014 Commun. Theor. Phys. 61 45 Google Scholar
[46]	Zhong W, Sun Z, Ma J, Wang X, Nori F 2013 Phys. Rev. A 87 022337 Google Scholar
[47]	Chruscinski D, Wudarski F 2013 Phys. Lett. A 377 1425 Google 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 140401 Google 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 062608 Google 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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