Entropy production versus memory effects in two-level open quantum systems

Problem Statement
The concept of entropy production is central to classical and quantum thermodynamics, linking irreversibility, information loss, and the arrow of time. Recent literature has suggested a close relationship between entropy production and memory effects (non-Markovianity). However, a significant challenge arises in the quantum regime: the non-unique characterization of non-Markovianity and the existence of multiple, often conflicting, definitions of entropy production. While some definitions coincide in the weak coupling limit, it is unclear how they behave in strong coupling regimes or how they relate to memory effects in those regimes. Specifically, it remains an open question whether negative entropy production rates universally imply non-Markovianity and which definition of entropy production is most appropriate for analyzing these phenomena in finite baths and strong coupling.

Methodology
The authors investigate these issues using a minimal but experimentally relevant model: a two-level system (qubit) interacting with a single bosonic mode acting as a finite bath. The interaction is modeled via the Jaynes-Cummings Hamiltonian. The study employs the following methodological steps:

1. Comparison of Definitions: The paper systematically compares five distinct definitions of entropy production rate:

   • $\sigma_{Es}$ (Esposito): Based on system entropy change and bath energy change.
   • $\sigma_{El}$ (Elouard): Based on system entropy change and the variation of the bath's "thermal energy."
   • $\sigma_{Co}$ (Correlation-based): Derived from an exact master equation involving an effective Hamiltonian.
   • $\sigma_{fp}$ (Fixed Point): Based on the relative entropy between the system state and the instantaneous fixed point of the dynamical map.
   • $\dot{I}_{A:B}$: The rate of mutual information buildup between the system and the bath.

2. Numerical Analysis: The authors simulate the time evolution of these rates under both weak and strong coupling regimes. They analyze the behavior of these rates for various initial states of the qubit.

3. Memory Effect Characterization: The study characterizes memory effects using P-divisibility (positive divisibility) of the dynamical map, as well as CP-divisibility and the BLP criterion. P-divisibility is determined by the time-dependent damping rates ($\gamma_1, \gamma_2, \gamma_3$) of the master equation.

4. Analytical Extension: To address discrepancies found in the strong coupling regime, the authors introduce a "map entropy production" ($\sigma_{map}$), defined as the minimum entropy production over all possible initial states at a given instant, dependent only on the instantaneous generator of the dynamics. They provide an analytical proof of the equivalence between the non-negativity of $\sigma_{map}$ and P-divisibility for phase-covariant master equations.

Key Results

• Weak Coupling Regime: In the weak coupling limit, all definitions of entropy production ($\sigma_{Es}, \sigma_{El}, \sigma_{Co}, \sigma_{fp}, \dot{I}_{A:B}$) coincide almost exactly. Furthermore, there is a high degree of correspondence between the sign of these rates and P-divisibility: whenever the dynamics is P-divisible, the entropy production is non-negative for all initial states. If the dynamics is not P-divisible, there exists at least one initial state for which the entropy production is negative.

• Strong Coupling Regime: Significant discrepancies emerge between the different definitions of entropy production. However, two specific definitions, $\sigma_{Es}$ (based on local system and bath quantities) and $\sigma_{fp}$ (based on the non-local fixed point of the map), are found to coincide exactly, even in the strong coupling regime. This equivalence is proven analytically in the appendices.

• Breakdown of Standard Correspondence: In the strong coupling regime, the direct correspondence between the sign of the standard entropy production rates (specifically the minimum over initial states, $\sigma_{min}$) and P-divisibility degrades. There are instances where the dynamics is not P-divisible, yet the minimum entropy production remains positive.

• Map Entropy Production: To restore the link between memory effects and entropy production in the strong coupling regime, the authors propose $\sigma_{map}$. They demonstrate both numerically and analytically that the sign of $\sigma_{map}$ is perfectly equivalent to P-divisibility. Specifically, $\sigma_{map}(t) \geq 0$ if and only if the dynamics is P-divisible at time $t$.

Significance and Claims
The paper claims to provide a broad physical interpretation of various entropy production definitions and the concept of P-divisibility. Its primary contributions are:

1. Clarification of Definitions: It establishes that while various entropy production definitions diverge in strong coupling, the Esposito and fixed-point definitions are exactly equivalent, suggesting a robust underlying physical quantity.
2. Refined Link to Memory Effects: The study demonstrates that the relationship between negative entropy production and non-Markovianity (memory effects) is not universal across all definitions or coupling regimes.
3. Extension to Dynamical Maps: By extending the concept of entropy production to the dynamical map level ($\sigma_{map}$), the authors establish a perfect equivalence between the sign of this quantity and P-divisibility. This resolves the ambiguity found in strong coupling regimes where standard system-level entropy production rates fail to fully capture the memory effects characterized by P-divisibility.

The authors conclude that their study establishes a direct equivalence between the presence of memory effects and the sign of an entropy-production-related quantity, provided the concept is appropriately extended to the dynamical map for strong coupling scenarios.