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arXiv:2501.17660 · Physical Review Research

Entropic witness for quantum memory in open system dynamics

Charlotte Bäcker, Konstantin Beyer, Walter T. Strunz

Published Jul 21, 2025• 3 citations
arXivHTMLPDF

Abstract

The dynamics of open quantum systems are often modeled by non-Markovian processes that account for memory effects arising from interactions with the environment. It is well known that the memory provided by the environment can be classical or quantum in nature. Remarkably, the quantumness of the memory can be witnessed locally by measurements on the open system alone, without requiring access to the environment. However, existing witnesses are computationally challenging for systems beyond qubits, which severely limits their practical applicability. In this work, we present a tractable criterion for quantum memory based on the von Neumann entropy, which is easily computable for systems of any dimension. Using this witness, we investigate the nature of memory in a class of physically motivated finite-dimensional qudit dynamics. Moreover, we demonstrate that this criterion is also suitable for detecting quantum memory in continuous-variable systems. As an illustrative example, we analyze non-Markovian Gaussian dynamics of a damped harmonic oscillator.

DOI
10.1103/618n-fp8w

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The dynamics of open quantum systems are often modeled by non-Markovian processes that account for memory effects arising from interactions with the environment. It is well known that the memory provided by the environment can be classical or quantum in nature. Remarkably, the quantumness of the memory can be witnessed locally by measurements on the open system alone, without requiring access to the environment. However, existing witnesses are computationally challenging for systems beyond qubits, which severely limits their practical applicability. In this work, we present a tractable criterion for quantum memory based on the von Neumann entropy, which is easily computable for systems of any dimension. Using this witness, we investigate the nature of memory in a class of physically motivated finite-dimensional qudit dynamics. Moreover, we demonstrate that this criterion is also suitable for detecting quantum memory in continuous-variable systems. As an illustrative example, we analyze non-Markovian Gaussian dynamics of a damped harmonic oscillator.
Non-Markovian processes may arise in physics due to memory effects of environmental degrees of freedom. For quantum non-Markovianity, it is an ongoing debate to clarify whether such memory effects have a verifiable quantum origin, or whether they might equally be modeled by a classical memory. In this contribution, we propose a criterion to test locally for a truly quantum memory. The approach is agnostic with respect to the environment, as it solely depends on the local dynamics of the system of interest. Experimental realizations are particularly easy, as only single-time measurements on the system itself have to be performed. We study memory in a variety of physically motivated examples, both for a time-discrete case, and for time-continuous dynamics. For the latter, we are able to provide an interesting class of non-Markovian master equations with classical memory that allows for a physically measurable quantum trajectory representation.
We present a method to detect quantum memory in a non-Markovian process. We call a process Markovian when the environment does not provide a memory that retains correlations across different system-environment interactions. We define two types of non-Markovian processes, depending on the required memory being classical or quantum. We formalise this distinction using the process matrix formalism, through which a process is represented as a multipartite state. Within this formalism, a test for entanglement in a state can be mapped to a test for quantum memory in the corresponding process. This allows us to apply separability criteria and entanglement witnesses to the detection of quantum memory. We demonstrate the method in a simple model where both system and environment are single interacting qubits and map the parameters that lead to quantum memory. As with entanglement witnesses, our method of witnessing quantum memory provides a versatile experimental tool for open quantum systems.
In the study of open quantum systems, information exchange between system and its surrounding environment plays an eminent and important role in analysing the dynamics of open quantum system. In this work, by making use of the quantum information theory and intrinsic properties such as \emph{entropy exchange}, \emph{coherent information} and using the notion of \emph{quantum loss} as a criterion of the amount of lost information, we will propose a new witness, based on information exchange, to detect non-Markovianity. Also a measure for determining the degree of non-Markovianity, will be introduced by using our witness. The characteristic of non-Markovianity is clarified by means of our witness, and we emphasize that this measure is constructed based on the loss of information or in other word the rate of \emph{quantum loss} in the environment. It is defined in term of reducing correlation between system and ancillary. Actually, our focus is on the information which be existed in the environment and it has been entered to the environment due to its interaction with the system. Remarkably, due to choosing the situation which the "system +ancillary" in maximal entangled pure state, optimization procedure does not need in calculation of our measure, such that the degree of non-Markovianity is computed analytically by straightforward calculations.
Accessing the physical mechanisms behind non-Markovian phenomena in open quantum dynamics requires the study of the statistical properties of the joint system-environment dynamics. This is impossible at the level of the reduced dynamics of the open system alone as the latter is obtained by suitably eliminating the environment. The task is instead made possible by considering multi-time correlation functions involving observables of the open system, only: the open system-environment interactions turn them into global ones thus building up correlations between the two systems. Multi-time correlations form the basis of both the theory of quantum stochastic processes and of the Alicki-Lindblad-Fannes dynamical entropy (ALF entropy for short). This latter quantity provides for quantum systems a measure of the dynamical entropy production as the Kolmogorov-Sinai entropy does for classical systems. In the case of a collisional model whereby the dissipative dynamics of a finite-level system is obtained by its coupling to an infinite classical spin chain, the ALF entropy can be explicitly computed. It turns out to depend on the parameters characterizing the statistical properties of the environment and can be related to the activation and super-activation of memory effects in the open quantum system.
The modeling and analysis of the dynamics of complex systems often requires to employ non-Markovian stochastic processes. While there is a clear and well-established mathematical definition for non-Markovianity in the case of classical systems, the extension to the quantum regime recently caused a vivid debate, leading to many different proposals for the characterization and quantification of memory effects in the dynamics of open quantum systems. Here, we derive a mathematical representation for the non-Markovianity measure based on the exchange of information between the open system and its environment, which reveals the locality and universality of non-Markovianity in the quantum state space and substantially simplifies its numerical and experimental determination. We further illustrate the application of this representation by means of an all-optical experiment which allows the measurement of the degree of memory effects in a photonic quantum process with high accuracy.
Non-Markovian effects due to quantum memory in the dynamics of open systems typically correspond to information backflows from the surrounding environment to the system. We propose a witness to quantify the non-Markovianity of quantum evolutions using the Bhattacharyya distance (BD), a specific quantum statistical distance. This witness has the advantage of not requiring the calculation of the evolved density matrix and only computes through the initial and final states of the system, therefore leading to the improvement of quantum metrology. It means that we calculate the quantum angle between two states to detect non-Markovian effects. This proposal is investigated by considering several instances of open quantum systems, such as two and three-level atoms interacting in single and two-mode fields, respectively, and two effective two-level atoms interacting locally with two independent environments. We demonstrate that the suggested BD-based non-Markovianity witness identifies memory effects, consistent with well-established witnesses based on Bures distance, quantum Fisher information, and Hilbert-Schmidt speed, showing sensitivity to information backflows.