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arXiv:2502.15498 · Physical Review A

Characterization of P-divisibility in two-level open quantum systems

Guillaume Théret, Camille Lombard Latune, Dominique Sugny

Published Oct 9, 2025• 1 citation
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Abstract

We study different characterizations of positive (P) divisibility in two-level open quantum systems whose dynamics are governed by a time-local master equation with time-dependent relaxation rates. In the case of phase-covariant dynamics, necessary and sufficient conditions for the P-divisibility of the dynamical map are given in terms of inequalities on such relaxation rates. We propose a geometric P-divisibility criterion based on the Bloch coordinates. The connection to the Breuer-Laine-Piilo (BLP) measure of non-Markovianity is also established. We show how these results can be extended to a Redfield master equation. As an application of such characterizations, we study the open dynamics of a qubit interacting with a single bosonic mode. More precisely, we characterize the properties of the local map of the qubit generated by its interaction with the bosonic mode, playing the role of an extremely reduced bath. Interesting observations are made, opening perspectives for a deeper physical understanding of complete positive divisibility, P-divisibility, and BLP measure.

DOI
10.1103/ngjb-mcnt

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We study different characterizations of positive (P) divisibility in two-level open quantum systems whose dynamics are governed by a time-local master equation with time-dependent relaxation rates. In the case of phase-covariant dynamics, necessary and sufficient conditions for the P-divisibility of the dynamical map are given in terms of inequalities on such relaxation rates. We propose a geometric P-divisibility criterion based on the Bloch coordinates. The connection to the Breuer-Laine-Piilo (BLP) measure of non-Markovianity is also established. We show how these results can be extended to a Redfield master equation. As an application of such characterizations, we study the open dynamics of a qubit interacting with a single bosonic mode. More precisely, we characterize the properties of the local map of the qubit generated by its interaction with the bosonic mode, playing the role of an extremely reduced bath. Interesting observations are made, opening perspectives for a deeper physical understanding of complete positive divisibility, P-divisibility, and BLP measure.
We study the concepts of complete positivity and positivity in a two-level open quantum system whose dynamics are governed by a time-local quantum master equation. We establish necessary and sufficient conditions on the time-dependent relaxation rates to ensure complete positivity and positivity of the dynamical map. We discuss their relations with the non-Markovian behavior of the open system. We also analyze the long-time asymptotic behavior of the dynamics as a function of the rates. We show under which conditions on the rates the system tends to an equilibrium state. Different examples illustrate this general study.
Divisibility of dynamical maps is a central notion in the study of quantum non-Markovianity, providing a natural framework to characterize memory effects via time-local master equations. In this work, we generalize the notion of divisibility of quantum dynamical maps from the Schrödinger to the Heisenberg picture. While the two pictures are equivalent at the level of physical predictions, we show that the divisibility properties of the corresponding dual maps are, in general, not equivalent. This inequivalence originates from the distinction between left and right generators of time-local master equations, which interchange roles under duality. We demonstrate that Schrödinger and Heisenberg divisibility are distinct concepts by constructing explicit dynamics divisible only in one picture. Furthermore, we introduce a quantifier for the violation of Heisenberg P-divisibility, analogous to the trace-distance-based measure of non-Markovianity, and provide it with an operational interpretation in terms of the guessing probability between effects. Our results show that Heisenberg divisibility is an independent witness of memory effects and highlight the need to consider both pictures when characterizing non-Markovian quantum dynamics.
Phase covariant qubit dynamics describes an evolution of a two-level system under simultaneous action of pure dephasing, energy dissipation, and energy gain with time-dependent rates γz(t)\gamma_{z}(t)γz​(t) , γ−(t)\gamma_{-}(t)γ−​(t) , and γ+(t)\gamma_{+}(t)γ+​(t) , respectively. Non-negative rates correspond to completely positive divisible dynamics, which can still exhibit such peculiarities as non-monotonicity of populations for any initial state. We find a set of quantum channels attainable in the completely positive divisible phase covariant dynamics and show that this set coincides with the set of channels attainable in semigroup phase covariant dynamics. We also construct new examples of eternally indivisible dynamics with γz(t)<0\gamma_{z}(t)<0γz​(t)<0 for all t>0t>0t>0 that is neither unital nor commutative. Using the quantum Sinkhorn theorem, we for the first time derive a restriction on the decoherence rates under which the dynamics is positive divisible, namely, γ±(t)≥0\gamma_{\pm}(t)\geq 0γ±​(t)≥0 , γ+(t)γ−(t)+2γz(t)>0\sqrt{\gamma_{+}(t)\gamma_{-}(t)}+2\gamma_{z}(t)>0γ+​(t)γ−​(t)​+2γz​(t)>0 . Finally, we consider phase covariant convolution master equations and find a class of admissible memory kernels that guarantee complete positivity of the dynamical map.
This technical note introduces clockability as an operational criterion for assessing the validity of time-local descriptions in open quantum systems. While non-Markovian dynamics are commonly characterized through information backflow or violations of CP-divisibility, the structural conditions under which a temporal description remains operationally meaningful are rarely made explicit. This work reframes non-Markovianity not as a dynamical anomaly, but as a signal of local clockability breakdown: the failure of the environment to function as a stable register of persistent and locally accessible traces. Within this framework, CP-divisibility emerges as a sufficient condition for local clockability, whereas its violation marks the loss of applicability of time-local master equations rather than a failure of quantum dynamics itself. The note clarifies the relationship between divisibility-based and information-theoretic measures of non-Markovianity and motivates a hierarchical domain-of-validity analysis for temporal descriptions. No new physical mechanisms or mathematical formalisms are introduced. Instead, the work provides a diagnostic, information-theoretic perspective on when time-local representations (e.g., GKSL forms) are structurally justified, and when dynamical descriptions should be interpreted as non-local, history-dependent responses. This contribution is intended as a conceptual and methodological clarification relevant to quantum information theory, open quantum systems, and the operational foundations of time.
We study a system qubit repeatedly interacting with the same environmental qubit, with a reservoir acting on the environment between collisions via a completely positive, trace-preserving map. We show that complete suppression of system--environment correlations uniquely requires a full environmental reset, recovering a semi group dynamics with a time-independent Gorini--Kossakowski--Sudarshan--Lindblad generator, whereas a partial reset yields a continuous transition between Markovian and non-Markovian regimes governed by a single dimensionless relaxation parameter. For a resonant excitation-exchange interaction, we obtain exact closed-form expressions for the Bloch-vector dynamics for both a generalized depolarizing channel and a generalized amplitude-damping channel acting as the reservoir-induced map. Using the Breuer--Laine--Piilo measure and a Choi-matrix CP-divisibility witness, we identify three distinct dynamical regimes across the parameter space: CP-divisible Markovian dynamics, CP-indivisible but P-divisible dynamics, and non-P-divisible non-Markovian dynamics. The boundaries between these regimes, and the structural differences between uniform and anisotropic environmental relaxation, are characterized numerically.
We compare several definitions of entropy production rate introduced in the literature from a large variety of situations and motivations, and then analyze their relations with memory effects. Considering a relevant experimental example of a qubit interacting with a single bosonic mode playing the role of a finite bath, we show that all definitions of entropy production coincide at weak coupling. In a stronger coupling regime, significant discrepancies emerge between the different entropy production rates, although some similarities in the overall behaviour remain. However, surprisingly, two of these definitions --one based on local quantities of the system and the other on non-local quantities-- coincide exactly, even in the case of strong coupling. Finally, a high degree of correspondence is observed when memory effects characterized by P-divisibility are compared with the sign of all entropy production rates in the case of weak coupling. Such correspondence degrades at stronger coupling, leading us to extend the concept of entropy production to the dynamical map. We show a perfect equivalence between the sign of this enlarged concept of entropy production and P-divisibility, both numerically and analytically, in the case of phase-covariant master equations.