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arXiv:2409.08749 · Physical Review A

Phase-space measures of information flow in open systems: A quantum and classical perspective of non-Markovianity

Moritz F. Richter, Heinz‐Peter Breuer

Published Dec 2, 2024• 2 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.110.062401

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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.
In this doctoral thesis we will investigate how one can quantify the information flow in quantum and classical phase space models based on the Breuer-Laine-Piilo measure of quantum non-Markovianity, quantifying memory effects of some open quantum dynamics by the information flow between the open system and its environment. While this mea- sure is defined using the trace distance between density operators on the Hilbert space, in case of continuous variable quantum systems it would be desirable to quantify the infor- mation flow by means of the differently ordered quasi-probability distributions over their quantum phase space. We will show how one can approximate the trace distance by the Kolmogorov distance between Glauber P-functions and Husimi Q-functions and we will prove the existence of some optimal ordering such that the Kolmogorov distance between the respectively ordered quasi-distributions equals the trace distance. The Kolmogorov dis- tance represents in this context a classical analogue to the trace distance and can be used in the spirit of the Breuer-Laine-Piilo measure to quantify information flow in classical phase space models. We hence define the limit of classical uncertainty which presents a quantum- to-classical transition in which the approximation of the trace distance by the Kolmogorov distances of the quasi-probability distributions becomes exact. Employing the example of the Calderia-Leggett model of quantum Brownian motion we present some numerical data which shows that the Kolmogorov distance between the Wigner functions converges espe- cially fast against the trace distance, strongly suggesting that the Wigner function in fact is the optimal ordered quasi-distribution in the limit of classical uncertainty. Additionally, or method can be used to compare the information flow through quantum systems and their classical counterparts. We do so exemplarily for classical and quantum anharmonic chains as used in the Fermi-Pasta-Ulam-Tsingou (FPUT) model. We can show, that the infor- mation flow through the quantum and the classical chain is equivalent in case of harmonic coupling between the sites of the chain while significant differences emerge for anharmonic coupling. We can relate this differences to the resolution of the phase space distributions, limited by the Heisenberg uncertainty relation in the quantum case and arbitrary small for the classical chain, and the role of entanglement as genuine quantum correlation type.
We introduce a non-Markovianity measure for continuous-variable open quantum systems based on the idea put forward in H.-P. Breuer et al. [Phys. Rev. Lett. 103, 210401 (2009);], that is, by quantifying the flow of information from the environment back to the open system. Instead of the trace distance we use here the fidelity to assess distinguishability of quantum states. We employ our measure to evaluate non-Markovianity of two paradigmatic Gaussian channels: the purely damping channel and the quantum Brownian motion channel with Ohmic environment. We consider different classes of Gaussian states and look for pairs of states maximizing the backflow of information. For coherent states we find simple analytical solutions, whereas for squeezed states we provide both exact numerical and approximate analytical solutions in the weak coupling limit.
Abstract We employ frames consisting of rank-one projectors (i.e. pure quantum states) and their induced informationally complete quantum measurements to represent generally mixed quantum states by quasi-probability distributions. In the case of discrete frames on finite dimensional systems this results in a vector like representation by quasi-probability vectors, while for the continuous frame of coherent states in continuous variable (CV) systems the approach directly leads to the celebrated representation by Glauber–Sudarshan P-functions and Husimi Q-functions. We explain that the Kolmogorov distances between these quasi-probability distributions lead to upper and lower bounds of the trace distance which measures the distinguishability of quantum states. We apply these results to the dynamics of open quantum systems and construct a non-Markovianity witness based on the Kolmogorov distance of the P- and Q-functions. By means of several examples we discuss the performance of this witness and demonstrate that it is useful in the regime of high entropy states for which a direct evaluation of the trace distance is typically very demanding. For Gaussian dynamics in CV systems we even find a suitable non-Markovianity measure based on the Kolmogorov distance between the P-functions which can alternatively be used as a witness for non-Gaussianity.
Characterization and quantification of non-Markovian dynamics in open quantum systems are topical issues in the rapidly developing field of quantum computation and quantum communication. A standard approach based on the notion of information backflow detects the flow of information from the environment back to the system. Numerous measures of information backflow have been proposed using different definitions of distinguishability between pairs of quantum states. These measures, however, necessitate optimization over the state space, which can be analytically challenging or numerically demanding. Here we propose an alternative witness and measure of information backflow that is explicitly state independent by utilizing the concept of quasiprobability representation and recent advances in the theory of majorization for quasiprobabilities. We illustrate its use over several paradigmatic examples, demonstrating consistent Markovian conditions with known results and also reported necessary and sufficient conditions for the qutrit system in a random unitary channel. The paper concludes with a discussion of the foundational implications of quantum dynamical evolution.
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.
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.