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

TL;DR

This work develops a direct link between Kolmogorov consistency violation and non-Markovian memory in open quantum dynamics, providing an operational bridge between temporal quantum correlations, information flow, and thermodynamics. By analyzing a dissipative single-qubit model with time-local master equations, the authors derive exact relations tying KC violations to CP-divisibility (RHP) and information-backflow (BLP) measures, mutual information, the Fano factor, heat exchange, and entropy production. They also connect KC violation to the Kirkwood–Dirac quasi-distribution and Leggett–Garg inequalities, revealing a common memory-driven mechanism—coherence revivals arising from temporarily negative decay rates. The results offer a unified framework to quantify quantum temporal non-classicality and identify thermodynamic and information-theoretic witnesses of non-Markovian dynamics with potential implications for quantum technologies and foundational questions.

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.

Figures (2)

• Figure 1: Analytical results for Case I. The spectral density exponent is fixed at $s=1.5$, the system--bath coupling strength at $\alpha=0.5$, and the bath temperature at $T=300$, while the system Hamiltonian frequency is $\omega_0=1.0$. To make every parameter dimensionless, $t'=\omega_0t$. The $\Gamma'(t)=\Gamma(t')/\omega_0$. (a) Time dependence of the decoherence rate $\Gamma(t)$ for different cutoff frequencies $\omega_c$. Varying $\omega_c$ significantly modifies the structure of $\Gamma(t)$, in particular the extent of its negative regions, thereby allowing control over the degree of non-Markovianity. (b) Total positive area $M(t)$ and total negative area $N(t)$ of the decay rate, evaluated over a long time window ($t=30$), as functions of $\omega_c$. The positive contribution dominates for all cutoff frequencies, leading to an overall increase of the net area, as highlighted in the inset. (c) Kolmogorov-consistency violation $\mathrm{viol}(t_1,t_2)$ as a function of $\omega_c$. The measurement is done at $t_1=15.0$ and the second at $t_2=30.0$. In agreement with the analytical expression Eq. \ref{['Eq:viol_MN']}, the violation decreases with increasing cutoff frequency, reflecting the reduction of the negative-rate contributions, as also illustrated in the inset.
• Figure 2: Analytical results for Case II. All system and bath parameters are identical to those used in Fig. \ref{['Fig:Gamma_Area_KCV']}(a), so the positive and negative areas of the decay rates coincide with those of Case I. The total evolution time and the measurement times are also chosen to be the same. The KC violation exhibits the same qualitative dependence on the positive and negative decay-rate areas, confirming that its behavior is governed by their interplay rather than by case-specific details.