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Quantum Markov monogamy of temporal correlations

Matheus Capela, Lucas C. Céleri, Rafael Chaves, Kavan Kishore Modi

Published Aug 26, 2021• 0 citations
OpenAlex

Abstract

Markovianity lies at the heart of classical communication problems. This in turn makes the information-theoretic characterization of Markovian processes worthwhile. Data processing inequalities are ubiquitous in this sense, assigning necessary conditions for all Markovian processes. We address here the problem of the information-theoretic analysis of constraints on Markovian processes in the quantum regime. Firstly, we show the existence of a novel class of quantum data processing inequalities called here quantum Markov monogamy inequalities. This new class of necessary conditions on quantum Markovian processes is inspired by its counterpart for classical Markovian processes, and thus providing a strong link between classical and quantum constraints on Markovianity. Secondly, we show the relevance of such inequalities by considering an example of non-Markovian behaviour witnessed by a monogamy inequality, nevertheless, do not violating any of the remaining data processing inequalities. Lastly, we show how this inequalities can be used to witness non-Markovianity at the level of the process tensor formalism.

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Markovianity lies at the heart of classical communication problems. This in turn makes the information-theoretic characterization of Markovian processes worthwhile. Data processing inequalities are ubiquitous in this sense, assigning necessary conditions for all Markovian processes. We address here the problem of the information-theoretic analysis of constraints on Markovian processes in the quantum regime. Firstly, we show the existence of a novel class of quantum data processing inequalities called here quantum Markov monogamy inequalities. This new class of necessary conditions on quantum Markovian processes is inspired by its counterpart for classical Markovian processes, and thus providing a strong link between classical and quantum constraints on Markovianity. Secondly, we show the relevance of such inequalities by considering an example of non-Markovian behaviour witnessed by a monogamy inequality, nevertheless, do not violating any of the remaining data processing inequalities. Lastly, we show how this inequalities can be used to witness non-Markovianity at the level of the process tensor formalism.
Markovianity lies at the heart of communication problems. This in turn makes the information-theoretic characterization of Markov processes worthwhile. Data-processing inequalities are ubiquitous in this sense, assigning necessary conditions for all Markov processes. We address here the problem of the information-theoretic analysis of constraints on Markov processes in the quantum regime. We show the existence of a class of quantum data-processing inequalities called here quantum Markov monogamy inequalities. This class of necessary conditions on quantum Markov processes is inspired by its counterpart for classical Markov processes, thus providing a strong link between classical and quantum constraints on Markovianity. We go on to construct a family of multitime quantum Markov monogamy inequalities, based on the process tensor formalism and that exploits multitime correlations. We then show, by means of an explicit example, that the Markov monogamy inequalities can be stronger than the usual quantum data-processing inequalities.
In the dynamics of open quantum systems, information may propagate in time through either the system or the environment, giving rise to Markovian and non-Markovian temporal correlations, respectively. However, despite their notable coexistence in most physical situations, it is not yet clear how these two quantities may limit the existence of one another. Here, we address this issue by deriving several inequalities relating the temporal correlations of general multitime quantum processes. The dynamics are described by process tensors, and the correlations are quantified by the mutual information between subsystems of their Choi states. First, we prove a set of upper bounds to the non-Markovianity of a process given the degree of Markovianity in each of its steps. This immediately implies a nontrivial maximum value for the non-Markovianity of any process, independent of its Markovianity. Finally, we determine how the non-Markovianity limits the amount of total temporal correlations that could be present in a given process. These results show that, although any multitime process must pay a price in total correlations to have a given amount of non-Markovianity, this price vanishes exponentially with the number of steps of the process, while the maximum non-Markovianity grows only linearly. This implies that even a highly non-Markovian process might be arbitrarily close to having the maximum amount of total correlations if it has a sufficiently large number of steps.
Understanding temporal processes and their correlations in time is of paramount importance for the development of near-term technologies that operate under realistic conditions. Capturing the complete multi-time statistics that define a stochastic process lies at the heart of any proper treatment of memory effects. This is well understood in classical theory, where a hierarchy of joint probability distributions completely characterizes the process at hand. However, attempting to generalize this notion to quantum mechanics is problematic: observing realizations of a quantum process necessarily disturbs the state of the system, breaking an implicit, and crucial, assumption in the classical setting. This issue can be overcome by separating the experimental interventions from the underlying process, enabling an unambiguous description of the process itself and accounting for all possible multi-time correlations for any choice of interrogating instruments. In this paper, using a novel framework for the characterization of quantum stochastic processes, we first solve the long standing question of unambiguously describing the memory length of a quantum processes. This is achieved by constructing a quantum Markov order condition, which naturally generalizes its classical counterpart for the quantification of finite-length memory effects. As measurements are inherently invasive in quantum mechanics, one has no choice but to define Markov order with respect to the interrogating instruments that are used to probe the process at hand: different memory effects are exhibited depending on how one addresses the system, in contrast to the standard classical setting. We then fully characterize the structural constraints imposed on quantum processes with finite Markov order, shedding light on a variety of memory effects that can arise through various examples. Finally, we introduce an instrument-specific notion of memory strength that allows for a meaningful quantification of the temporal correlations between the history and the future of a process for a given choice of experimental intervention. These findings are directly relevant to both characterizing and exploiting memory effects that persist for a finite duration. In particular, immediate applications range from developing efficient compression and recovery schemes for the description of quantum processes with memory to designing coherent control protocols that efficiently perform information-theoretic tasks, amongst a plethora of others.
We derive a necessary and sufficient condition for a quantum process to be Markovian which coincides with the classical one in the relevant limit. Our condition unifies all previously known definitions for quantum Markov processes by accounting for all potentially detectable memory effects. We then derive a family of measures of non-Markovianity with clear operational interpretations, such as the size of the memory required to simulate a process or the experimental falsifiability of a Markovian hypothesis.
We present a comprehensive and up-to-date review of the concept of quantum non-Markovianity, a central theme in the theory of open quantum systems. We introduce the concept of a quantum Markovian process as a generalization of the classical definition of Markovianity via the so-called divisibility property and relate this notion to the intuitive idea that links non-Markovianity with the persistence of memory effects. A detailed comparison with other definitions presented in the literature is provided. We then discuss several existing proposals to quantify the degree of non-Markovianity of quantum dynamics and to witness non-Markovian behavior, the latter providing sufficient conditions to detect deviations from strict Markovianity. Finally, we conclude by enumerating some timely open problems in the field and provide an outlook on possible research directions.