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arXiv:2601.01122

Non-Markovian and Thermodynamic Signatures in the Classicality Assessment via Kolmogorov Consistency

Arghya Maity, Ahana Ghoshal, Kelvin Onggadinata, Teck Seng Koh

quant-phSubmitted Jan 3, 2026• Announced Jan 6, 2026• 0 citations
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Abstract

The Kolmogorov consistency condition (KCC) defines the statistical boundary between classical and quantum dynamics. Its violation signifies the breakdown of a classical Markov description of temporal correlations. In this work, we establish a direct analytical connection between KCC violation and non-Markovianity in open quantum dynamics, revealing how memory effects manifest as departures from classical probabilistic consistency. Within a generic two-level open quantum system framework, we establish quantitative connections between the magnitude of KCC violation and key information-theoretic and thermodynamic quantities, such as mutual information, the Fano factor, heat exchange, and entropy production rate, thereby enabling a thermodynamic interpretation of temporal quantum correlations. Furthermore, we uncover formal correspondences between KCC violation, the Leggett-Garg inequality, and the negativity of the Kirkwood-Dirac quasi-distribution, identifying them as complementary witnesses of temporal quantum non-classicality. Our results thus provide a unified framework linking information-theoretic, thermodynamic, and temporal indicators of quantumness in open quantum systems.

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21 pages, 2 figures

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The Kolmogorov consistency condition (KCC) defines the statistical boundary between classical and quantum dynamics. Its violation signifies the breakdown of a classical Markov description of temporal correlations. In this work, we establish a direct analytical connection between KCC violation and non-Markovianity in open quantum dynamics, revealing how memory effects manifest as departures from classical probabilistic consistency. Within a generic two-level open quantum system framework, we establish quantitative connections between the magnitude of KCC violation and key information-theoretic and thermodynamic quantities, such as mutual information, the Fano factor, heat exchange, and entropy production rate, thereby enabling a thermodynamic interpretation of temporal quantum correlations. Furthermore, we uncover formal correspondences between KCC violation, the Leggett-Garg inequality, and the negativity of the Kirkwood-Dirac quasi-distribution, identifying them as complementary witnesses of temporal quantum non-classicality. Our results thus provide a unified framework linking information-theoretic, thermodynamic, and temporal indicators of quantumness in open quantum systems.
In the present Thesis we study the behavior of multi-time correlation functions and of thermodynamical quantities such as heat in open quantum systems undergoing an evolution generally affected by the presence of memory effects, i.e. non-Markovian. In the last decade, a large part of the scientific community in this field has dedicated its efforts to the understanding, precise definition and quantification of non-Markovianity in the quantum realm and now we have at our disposal several benchmark results and a plethora of different estimators that allow to determine the degree of non-Markovianity of a given dynamics. It comes therefore natural to investigate how other different dynamical quantities relate to such estimators also in order to understand the physical implications of memory effects on the statistics of observable quantities. In the first part of this work, a quantitative test of the violation of the so-called quantum regression theorem in presence of a non-Markovian dynamical regime is investigated. The quantum regression theorem represents a procedure that, whenever valid, allows to reconstruct two-time correlation functions of system's operators from the sole knowledge of the dynamics of mean values. It is worth stressing that two-time correlation functions are necessary in order to fully characterize the statistical properties of a quantum system, since they are able to catch aspects of the dynamics, such as fluorescence spectrum, in general not accessible looking at mean values. Despite their relevance however, obtaining two-time correlation functions often represents a formidable task, since the knowledge of the full "system+environment" dynamics is required, a generally too demanding request in the context of open quantum systems theory. The quantum regression theorem represents in this regard the easiest route to determine two-time correlation functions, this highlighting its importance. In this work we show that, in a pure-dephasing spin-boson model, the quantum regression theorem represents a stronger condition than non-Markovianity, in the sense that any presence of memory effects in the reduced dynamics inevitably results in violations to the former. These results have been published in [G.Guarnieri, A. Smirne, B. Vacchini, Phys. Rev. A 90, 022110 (2014)]. The second part of the Thesis is devoted to the characterization of heat ow at the microscopic level in open quantum systems, both finite and infinite dimensional. In particular we begin by studying the time behavior of its mean value in a non-Markovian dynamical regime, showing that, at variance with what happens in the Born-Markov semigroup limiting case, heat can backflow from the environment to the system. After providing a condition for the occurrence of such phenomenon and a measure for its amount for a given dynamics, the relationship with suitable non-Markovianity estimators is sought in two paradigmatic models, namely the spin-boson and the quantum brownian motion. The results, collected in [ G. Guarnieri, C. Uchiyama, B. Vacchini, Phys. Rev. A 93, 012118 (2016); G. Guarnieri, J. Nokkala, R. Schmidt, S. Maniscalco, B. Vacchini, Phys. Rev. A 94, 062101 (2016)], on the one hand allow for the identification of parameter-regions where the heat backflow is absent or maximum. On the other hand they show that the occurrence of heat backflow represents a stricter condition than non-Markovianity, in the sense that non-Markovianity allows for the observation of heat flowing back from the environment to the system and, vice versa, a Markovian dynamics prevents its occurrence. This Thesis concludes with the formulation of a new family of lower bounds to the mean dissipated heat in an environmental-assisted erasure-protocol scenario where Landauer's principle applies. As originally conceived for classical systems, this principle states that every irreversible erasure of information stored in a system inevitably carries along an amount of heat dissipated into the environment which is expended to perform the action. Within the framework recently put forward in [D. Reeb, M. M. Wolf, New J. Phys. 16, 103011 (2014)], which guarantees the validity of Landauer's principle in an open quantum systems scenario, we provide an asymptotically tight family of lower bounds to the dissipated heat which are also valid in the non-equilibrium setting. This construction is applied to an open system consisting of a three-level V-system, in which one transition is externally pumped by a laser field while the other is coupled through an XX-interaction to an environment consisting of a spin chain. Beside calculating all these quantities, an exact solution for the dynamics of such system is also provided. These results are collected in [G. Guarnieri, S. Campbell, J. Goold, S. Pigeon, M. Paternostro, B. Vacchini, in preparation].
Quantum thermodynamics has emerged as a central field for understanding how energy conversion processes occur in microscopic systems. In these systems, effects such as coherence, entanglement, and non-Markovianity play key roles. In this thesis, we explore different ways to describe quantum thermodynamics, using two main approaches: one based on entropy and the other on ergotropy. First, we introduce a generalized approach to quantify non-Markovianity through the breakdown of monotonicity in thermodynamic functions. In this context, the entropy-based heat flow serves as a practical tool to witness and measure quantum memory in unital maps that do not reverse the sign of the internal energy. Next, we analyze the dynamics of ergotropy in open qubits under both Markovian and non-Markovian evolutions. We identify phenomena such as freezing and sudden death of ergotropy, and we establish an analytical relation between the change in ergotropy and the environment-induced work . This provides a clear physical interpretation for the additional term in the first law in the entropy-based formulation. Finally, we propose an ergotropy-based thermodynamic formulation, in which heat is reinterpreted in terms of the change of the passive state associated with the density operator governing the quantum dynamics. This approach allows one to measure non-Markovianity of unital maps more generally and accurately than the entropy-based heat flow. This advantage comes from the direct link between heat and von Neumann entropy, a property ensured by invariance under passive transformations. Moreover, the out-of-equilibrium temperature naturally remains non-negative, similarly to equilibrium thermodynamics.
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.
Abstract We investigate the nonclassicality of an open quantum system using Leggett–Garg inequalities (LGI) which test the correlations of a single system measured at different times. Violation of LGI implies nonclassical behavior of the open system. We investigate the violation of the Leggett–Garg inequality for a two level system (qubit) spontaneously decaying under a general non-Markovian dissipative environment. A robust nonclassical behavior is shown through the violation of LGI in the non-Markovian regime of the environment. Our results are exact as we have evaluated the two-time correlation functions exactly without using quantum regression theorem. We probe quantum signatures of the open system through the exact dynamics of Leggett–Garg inequalities for a wide range of system-environment parameters beyond Born–Markov regime.
We study the thermalization dynamics and thermodynamics of a qubit coupled to a structured, memory-carrying environment modeled as a single bosonic mode continuously thermalized by a Markovian bath. The competition between coherent qubit-mode coupling g and thermalization rate κ is controlled by the dimensionless ratio μ=g/κ, which drives a crossover between Markovian and strongly non-Markovian regimes. In the strongly non-Markovian limit, coherent energy exchange produces transient shortcuts to local thermalization, whereby the qubit approaches its thermal state long before global equilibration. In the opposite limit, we derive an original convolution master equation with an exponential memory kernel, which reduces to a thermal GKSL equation as μ→0. The crossover between the two regimes closely mirrors the underdamped-to-overdamped transition of a classical damped oscillator. From the thermodynamic side, we derive an original decomposition of the reduced entropy production in terms of qubit-ancilla mutual information and ancilla relative entropy, providing a quantitative fingerprint of non-Markovian corrections. We then extend this framework to the stochastic level via two monitoring protocols — Fock-state tracking and energy-exchange detection — constructing trajectory-level entropy productions and establishing fluctuation theorems tailored to each scheme. Together, these results demonstrate that a self-consistent stochastic thermodynamics can be formulated for non-Markovian open quantum systems under continuous monitoring.
We address the problem of identifying non-Markovian quantum time evolutions of an open quantum system by only performing measurements of the system's energy. We demonstrate that violations of CP-divisibility are always witnessed by non-positive values of the energy-change Kirkwood-Dirac quasiprobability distribution associated with the system's Hamiltonian, evaluated at consecutive times. The link between non CP-divisibility and non-positivity of the system's energy-change distribution is stronger when the system-environment interactions are energy-preserving. The witness works whenever anomalous energy fluxes, due to non-Markovianity, are realized. Anomalous fluxes are also detected by the non-Markovianity measure built over the quantum mutual information between the states of the open system and of a quantum correlated reference.