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Classical and quantum phase-space measures for information flow and non-markovianity

Moritz F. Richter

Published Jun 22, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.6094/unifr/284055

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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.
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.
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.
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.
We study quantum non-Markovian dynamics of the Caldeira-Leggett model, a prototypical model for quantum Brownian motion describing a harmonic oscillator linearly coupled to a reservoir of harmonic oscillators. Employing the exact analytical solution of this model one can determine the size of memory effects for arbitrary couplings, temperatures, and frequency cutoffs. Here, quantum non-Markovianity is defined in terms of the flow of information between the open system and its environment, which is quantified through the Bures metric as distance measure for quantum states. This approach allows us to discuss quantum memory effects in the whole range from weak to strong dissipation for arbitrary Gaussian initial states. A comparison of our results with the corresponding results for the spin-boson problem show a remarkable similarity in the structure of non-Markovian behavior of the two paradigmatic models.
We discuss the conceptually different definitions used for the non-Markovianity of classical and quantum processes. The well-established definition for non-Markovianity of a classical stochastic process represents a condition on the Kolmogorov hierarchy of the n-point joint probability distributions. Since this definition cannot be transferred to the quantum regime, quantum non-Markovianity has recently been defined and quantified in terms of the underlying quantum dynamical map, using either its divisibility properties or the behavior of the trace distance between pairs of initial states. Here, we investigate and compare these definitions and their relations to the classical notion of non-Markovianity by employing a large class of non-Markovian processes, known as semi-Markov processes, which admit a natural extension to the quantum case. A number of specific physical examples is constructed which allow to study the basic features of the classical and the quantum definitions and to evaluate explicitly the measures for quantum non-Markovianity. Our results clearly demonstrate several fundamental distinctions between the classical and the quantum notion of non-Markovianity, as well as between the various quantum measures for non-Markovianity.
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.