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arXiv:1804.06445 · Physical Review A

Generalized trace distance approach to quantum non-Markovianity and detection of initial correlations

Giulio Amato, Heinz‐Peter Breuer, Bassano Vacchini

Published Jul 17, 2018• 27 citations
arXivHTMLPDF

Abstract

A measure of quantum non-Markovianity for an open system dynamics, based on revivals of the distinguishability between system states, has been introduced in the literature using the trace distance as quantifier for distinguishability. Recently it has been suggested to use as measure for the distinguishability of quantum states the trace norm of Helstrom matrices, given by weighted differences of statistical operators. Here we show that this approach, which generalizes the original one, is consistent with the interpretation of information flow between the system and its environment associated to the original definition. To this aim we prove a bound on the growth of the external information, that is information which cannot be accessed by performing measurements on the system only, as quantified by means of the Helstrom matrix. We further demonstrate by means of example that it is of relevance in generalizing schemes for the local detection of initial correlations based on the increase of internal information. Finally we exploit this viewpoint to show the optimality of a previously introduced strategy for the local detection of quantum correlations.

DOI
10.1103/physreva.98.012120

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A measure of quantum non-Markovianity for an open system dynamics, based on revivals of the distinguishability between system states, has been introduced in the literature using the trace distance as quantifier for distinguishability. Recently it has been suggested to use as measure for the distinguishability of quantum states the trace norm of Helstrom matrices, given by weighted differences of statistical operators. Here we show that this approach, which generalizes the original one, is consistent with the interpretation of information flow between the system and its environment associated to the original definition. To this aim we prove a bound on the growth of the external information, that is information which cannot be accessed by performing measurements on the system only, as quantified by means of the Helstrom matrix. We further demonstrate by means of example that it is of relevance in generalizing schemes for the local detection of initial correlations based on the increase of internal information. Finally we exploit this viewpoint to show the optimality of a previously introduced strategy for the local detection of quantum correlations.
We establish a direct connection of quantum Markovianity of an open system to its classical counterpart by generalizing the criterion based on the information flow. Here the flow is characterized by the time evolution of Helstrom matrices, given by the weighted difference of statistical operators, under the action of the quantum dynamical map. It turns out that the introduced criterion is equivalent to PPP divisibility of a quantum process, namely, divisibility in terms of positive maps, which provides a direct connection to classical Markovian stochastic processes. Moreover, it is shown that mathematical representations similar to those found for the original trace-distance-based measure hold true for the associated generalized measure for quantum non-Markovianity. That is, we prove orthogonality of optimal states showing a maximal information backflow and establish a local and universal representation of the measure. We illustrate some properties of the generalized criterion by means of examples.
We analyze and compare different measures for the degree of non-Markovianity in the dynamics of open quantum systems. These measures are based on the distinguishability of quantum states, which is quantified, on the one hand, by the trace distance or, more generally, by the trace norm of the Helstrom matrix and, on the other hand, by entropic quantifiers: the Jensen-Shannon divergence and the Holevo or quantum skew divergence. We explicitly construct a qubit dynamics for which the trace-norm-based non-Markovianity measure is nonzero, while all the entropic measures turn out to be zero. This leads to the surprising conclusion that the non-Markovianity measure which employs the trace norm of the Helstrom matrix is strictly stronger than all entropic non-Markovianity measures.
We construct a general measure for the degree of non-Markovian behavior in open quantum systems. This measure is based on the trace distance which quantifies the distinguishability of quantum states. It represents a functional of the dynamical map describing the time evolution of physical states, and can be interpreted in terms of the information flow between the open system and its environment. The measure takes on nonzero values whenever there is a flow of information from the environment back to the open system, which is the key feature of non-Markovian dynamics.
The exchange of information between an open quantum system and its environment, especially the backflow of information from the environment to the open system associated with quantum notions of non-Markovianity, is a widely discussed topic for years now. This information flow can be quantified by means of the trace distance of pairs of quantum states which provides a measure for the distinguishability of the states. The same idea can also be used to characterize the information flow in classical open systems through a suitable distance measure for their probability distributions on phase space. Here, we investigate the connection between the trace distance based quantum measure and the Kolmogorov distance for differently ordered quasiprobability distributions on phase space. In particular, we show that, for any pair of quantum states, one can find a unique quasiprobability distribution for which the Kolmogorov distance coincides with the trace distance. We further study the quantum-to-classical transition of the distance measures. By employing the Caldeira-Leggett model of quantum Brownian motion as a prototypical example, numerical simulations indicate a particularly rapid convergence of the Kolmogorov distance of the Wigner functions to the trace distance in the classical uncertainty limit, which establishes the Wigner function distance as an optimal tool for measuring semi-classical information backflow and for quantifying non-Markovianity in open continuous variable quantum systems.
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.
A quantum system inevitably interacts with its surroundings. In general, one does not have detailed information on an environment. Identifying the environmental features can help us to control the environment and its effects on the dynamics of an open system. Here, we consider a tripartite system and introduce a witness for the initial correlations among environments by means of the concept of the trace distance. Due to the existence of the initial environmental correlations, a tight upper bound is obtained for the growth of the trace distance of open quantum system states. Therefore, the initial correlations among the environments subject to particular conditions can be detected by measurements on the open system.