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Physical Review A

Dynamics of Rényi entropy and applications in detecting quantum non-Markovianity

Hongting Song, Yuanyuan Mao

Published Sep 19, 2017• 10 citations
Publisher

Abstract

Exploiting the master equations in the Lindblad form, we establish a sufficient and necessary condition for a Markovian dynamics to be unital. Based on this condition, we analyze the dynamical property of quantum R\'enyi entropy and propose a characterization of quantum non-Markovianity for unital dynamics in terms of R\'enyi entropy, which contains the previous criteria of non-Markovianity via von Neumann entropy and linear entropy as particular cases. The effectiveness of this measure in capturing the backflow of information from the environment is illustrated through several typical unital dynamics.

DOI
10.1103/physreva.96.032115

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Exploiting the master equations in the Lindblad form, we establish a sufficient and necessary condition for a Markovian dynamics to be unital. Based on this condition, we analyze the dynamical property of quantum R'enyi entropy and propose a characterization of quantum non-Markovianity for unital dynamics in terms of R'enyi entropy, which contains the previous criteria of non-Markovianity via von Neumann entropy and linear entropy as particular cases. The effectiveness of this measure in capturing the backflow of information from the environment is illustrated through several typical unital dynamics.
The R\'enyi entropies constitute a family of information measures that generalizes the well-known Shannon entropy, inheriting many of its properties. They appear in the form of unconditional and conditional entropies, relative entropies or mutual information, and have found many applications in information theory and beyond. Various generalizations of R\'enyi entropies to the quantum setting have been proposed, most notably Petz's quasi-entropies and Renner's conditional min-, max- and collision entropy. Here, we argue that previous quantum extensions are incompatible and thus unsatisfactory. We propose a new quantum generalization of the family of R\'enyi entropies and the corresponding relative entropies that contains the min-entropy, collision entropy, von Neumann entropy as well as the max-entropy as special cases, thus encompassing most quantum entropies in use today. We show several natural properties for this definition, including data-processing inequalities and a limited duality relation. We conclude that the treatment of these entropies is technically challenging and requires sophisticated tools from linear algebra. We share several conjectured properties which we were unable to prove.
Master equations govern the time evolution of a quantum system interacting with an environment, and may be written in a variety of forms. Time-independent or memoryless master equations, in particular, can be cast in the well-known Lindblad form. Any time-local master equation, Markovian or non-Markovian, may in fact also be written in a Lindblad-like form. A diagonalization procedure results in a unique, and in this sense canonical, representation of the equation, which may be used to fully characterize the non-Markovianity of the time evolution. Recently, several different measures of non-Markovianity have been presented which reflect, to varying degrees, the appearance of negative decoherence rates in the Lindblad-like form of the master equation. We therefore propose using the negative decoherence rates themselves, as they appear in the canonical form of the master equation, to completely characterize non-Markovianity. The advantages of this are especially apparent when more than one decoherence channel is present. We show that a measure proposed by Rivas et al. [Phys. Rev. Lett. 105, 050403 (2010)] is a surprisingly simple function of the canonical decoherence rates, and give an example of a master equation that is non-Markovian for all times t>0t>0t>0, but to which nearly all proposed measures are blind. We also give necessary and sufficient conditions for trace distance and volume measures to witness non-Markovianity, in terms of the Bloch damping matrix.
Abstract A non-Markovianity measure based on Brukner–Zeilinger invariant information to characterize non-Markovian effect of open systems undergoing unital dynamical maps is proposed. The method takes advantage of non-increasing property of the Brukner–Zeilinger invariant information under completely positive and trace-preserving unital maps. The simplicity of computing the Brukner–Zeilinger invariant information is the advantage of the proposed measure because of mainly depending on the purity of quantum state. The measure effectively captures the characteristics of non-Markovianity of unital dynamical maps. As some concrete application, we consider two typical non-Markovian noise channels, i.e., the phase damping channel and the random unitary channel to show the sensitivity of the proposed measure. By investigation, we find that the conditions of detecting the non-Markovianity for the phase damping channel are consistent with the results of existing measures for non-Markovianity, i.e., information flow, divisibility and quantum mutual information. However, for the random unitary channel non-Markovian conditions are same to that of the information flow, but is different from that of the divisibility and quantum mutual information.
We introduce a generalized approach to characterize the non-Markovianity of quantum dynamical maps via breakdown of monotonicity of thermodynamic functions. By adopting an entropy-based formulation of quantum thermodynamics, we use the relationship between heat and entropy to propose a measure of non-Markovianity based on the heat flow for single-qubit quantum evolutions. This measure can be applied for unital dynamical maps that do not invert the sign of the internal energy. Under certain conditions, it can also be extended for other thermodynamic functions, such as internal energy and work flows. In this context, a natural connection between heat and quantum coherence can be identified for dynamical maps that are both unital and incoherent. As applications, we explore dissipative and nondissipative quantum dynamical processes, illustrating the compatibility between our thermodynamic quantifiers and the well-establish measure defined via quantum coherence.
The Rényi entropies constitute a family of information measures that generalizes the well-known Shannon entropy, inheriting many of its properties. They appear in the form of unconditional and conditional entropies, relative entropies, or mutual information, and have found many applications in information theory and beyond. Various generalizations of Rényi entropies to the quantum setting have been proposed, most prominently Petz's quasi-entropies and Renner's conditional min-, max-, and collision entropy. However, these quantum extensions are incompatible and thus unsatisfactory. We propose a new quantum generalization of the family of Rényi entropies that contains the von Neumann entropy, min-entropy, collision entropy, and the max-entropy as special cases, thus encompassing most quantum entropies in use today. We show several natural properties for this definition, including data-processing inequalities, a duality relation, and an entropic uncertainty relation.