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Physical Review A

Measuring non-Markovianity based on local quantum uncertainty

Zhi He, Chunmei Yao, Qiong Wang, Jian Zou

Published Oct 1, 2014• 50 citations
Publisher

Abstract

We introduce an alternative method based on local quantum uncertainty to measure non-Markovianity for an open quantum system. The local quantum uncertainty proposed by Girolami $et al$. [Phys. Rev. Lett. 110, 240402 (2013)] is defined by the minimum skew information from a local measurement perspective. The measure effectively captures the characteristics of non-Markovianity and has a clearly physical interpretation: in correlated bipartite systems, the non-Markovian (Markovian) process of a considered open subsystem corresponds to the increase (decrease) of quantum uncertainty of local observables of another measured free subsystem. Then the local quantum uncertainty is further discussed by carefully handling the square root of the density operator for the bipartite $2\ensuremath{\bigotimes}d$ systems. Specifically, for a two-qubit system we give an explicit expression of the local quantum uncertainty. Furthermore, by applying the measure to three typical non-Markovian noise channels, we find that the conditions of detecting the non-Markovianity are consistent with the results of existing measures for non-Markovianity, i.e., information backflow and divisibility.

DOI
10.1103/physreva.90.042101

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We introduce an alternative method based on local quantum uncertainty to measure non-Markovianity for an open quantum system. The local quantum uncertainty proposed by Girolami etalet aletal. [Phys. Rev. Lett. 110, 240402 (2013)] is defined by the minimum skew information from a local measurement perspective. The measure effectively captures the characteristics of non-Markovianity and has a clearly physical interpretation: in correlated bipartite systems, the non-Markovian (Markovian) process of a considered open subsystem corresponds to the increase (decrease) of quantum uncertainty of local observables of another measured free subsystem. Then the local quantum uncertainty is further discussed by carefully handling the square root of the density operator for the bipartite 2\ensuremath⨂d2\ensuremath{\bigotimes}d2\ensuremath⨂d systems. Specifically, for a two-qubit system we give an explicit expression of the local quantum uncertainty. Furthermore, by applying the measure to three typical non-Markovian noise channels, we find that the conditions of detecting the non-Markovianity are consistent with the results of existing measures for non-Markovianity, i.e., information backflow and divisibility.
Although a number of measures for quantum non-Markovianity have been proposed recently, it is still an open question whether these measures directly characterize the memory effect of the environment, i.e., the dependence of a quantum state on its past in a time evolution. In this paper, we present a criterion and propose a measure for non-Markovianity with clear physical interpretations of the memory effect. The non-Markovianity is defined by the inequality $T({t}_{2},{t}_{0})\ensuremath{\ne}T({t}_{2},{t}_{1})T({t}_{1},{t}_{0})$ in terms of memoryless dynamical map TTT introduced in this paper. This definition is conceptually distinct from that based on divisibility used by Rivas et al. [Phys. Rev. Lett. 105, 050403 (2010)], whose violation is manifested by noncomplete positivity of the dynamical map. We demonstrate via a typical quantum process that without Markovian approximation, nonzero memory effects (non-Markovianity) always exist even if the non-Markovianity is zero by the other non-Markovianity measures.
In the study of open quantum systems, memory effects are usually ignored, and this leads to dynamical semigroups and Markovian dynamics. However, in practice, non-Markovian dynamics is the rule rather than the exception. With the recent emergence of quantum information theory, there is a flurry of investigations of non-Markovian dynamics, and several significant measures for non-Markovianity are introduced from various perspectives such as deviation from divisibility, information exchange between a system and its environment, or entanglement with the environment. In this work, by exploiting the correlations flow between a system and an arbitrary ancillary, we propose a considerably intuitive measure for non-Markovianity by use of correlations as quantified by the quantum mutual information rather than entanglement. The fundamental properties, physical significance, and differences and relations with existing measures for non-Markovianity are elucidated. The measure captures quite directly and deeply the characteristics of non-Markovianity from the perspective of information. A simplified version based on Jamio\l{}kowski-Choi isomorphism which encodes operations via bipartite states and does not involve any optimization is also proposed.
Characterization of non-Markovian open quantum dynamics is both of theoretical and practical relevance. In a seminal work [Phys. Rev. Lett. 120, 040405 (2018)], a necessary and sufficient quantum Markov condition is proposed, with a clear operational interpretation and correspondence with the classical limit. Here we propose two non-Markovianity measures for general open quantum dynamics, which are fully reconciled with the Markovian limit and can be efficiently calculated based on the multi-time quantum measurements of the system. A heuristic algorithm for reconstructing the underlying open quantum dynamics is proposed, whose complexity is directly related to the proposed non-Markovianity measures. The non-Markovianity measures and the reconstruction algorithm are demonstrated with numerical examples, together with a careful reexamination of the non-Markovianity in quantum dephasing dynamics.
Abstract Based on the nonincreasing property of quantum coherence via skew information under incoherent completely positive and trace-preserving maps, we propose a non-Markovianity measure for open quantum processes. As applications, by applying the proposed measure to some typical noisy channels, we find that it is equivalent to the three previous measures of non-Markovianity for phase damping and amplitude damping channels, i.e. the measures based on the quantum trace distance, dynamical divisibility, and quantum mutual information. For the random unitary channel, it is equivalent to the non-Markovianity measure based on l 1 norm of coherence for a class of output states and it is incompletely equivalent to the measure based on dynamical divisibility. We also use the modified Tsallis relative entropy of coherence to detect the non-Markovianity of dynamics of quantum open systems, the results show that the modified Tsallis relative entropy of coherence are more comfortable than the original Tsallis relative entropy of coherence for small .
Recently, a measure for the non-Markovian behavior of quantum processes in open systems has been developed, which is based on the quantification of the flow of information between the open system and its environment [Phys. Rev. Lett. 103, 210401 (2009)]. The information flow is connected to the rate of change of the trace distance between quantum states, which can be interpreted in terms of the distinguishability of these states. Here, we elaborate the mathematical details of this theory, present applications to specific physical models, and discuss further theoretical and experimental implications as well as relations to alternative approaches proposed recently.
The quantification and characterization of non-Markovian dynamics in quantum systems is an essential endeavor both for the theory of open quantum systems and for a deeper understanding of the effects of non-Markovian noise on quantum technologies. Here, we introduce the robustness of non-Markovianity, an operationally-motivated, optimization-free measure that quantifies the minimum amount of Markovian noise that can be mixed with a non-Markovian evolution before it becomes Markovian. We show that this quantity is a bonafide non-Markovianity measure, since it is faithful, convex, and monotonic under composition with Markovian channels. A two-fold operational interpretation of this measure is provided, with the robustness measure quantifying an advantage in both a state discrimination and a channel discrimination task. Furthermore, we provide a closed-form analytical expression for this measure and show that, quite remarkably, the robustness measure is exactly equal to half the Rivas-Huelga-Plenio (RHP) measure [Phys. Rev. Lett. \textbf{105}, 050403 (2010)]. As a result, we provide a direct operational meaning to the RHP measure while endowing the robustness measure with the physical characterizations of the RHP measure.