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

TL;DR

This work compares diverse entropy-production definitions in open quantum dynamics using a qubit coupled to a single bosonic mode as a finite bath. It finds that all definitions agree at weak coupling but diverge under strong coupling, though two definitions ($\sigma^{Es}$ and $\sigma^{fp}$) coincide exactly and another pair ($\sigma^{El}$ and $\dot I_{A:B}$) remain closely aligned; it introduces a map-level entropy production $\sigma_{\text{map}}$ whose sign matches P-divisibility, with a rigorous proof for phase-covariant dynamics. The results establish a direct link between irreversibility and memory effects, showing that memory properties can be captured by map-level entropy production and highlighting the role of non-Markovian criteria in finite-bath thermodynamics. These findings illuminate how memory effects influence entropy production, offering a framework that could guide control and thermodynamic tasks in quantum technologies with small environments.

Abstract

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 the strong 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 strong 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.

Figures (11)

• Figure 1: Plots of the different definitions of entropy production $\sigma^{Es}$ (orange, thick), $\sigma^{El}$ (green), $\sigma^{Co}$ (purple), $\sigma^{fp}$ (dashed blue), and $\dot I_{A:B}$ (dot-dashed black) in a weak coupling regime, with $\omega_B/\omega_A = 0.6$, $\Delta/\omega_A = 0.4$, $\omega_A\beta_A = 1.1$, $\omega_A\beta_B = 0.3$, and $g/\omega_A =0.03$.
• Figure 2: Plots of the different definitions of entropy production $\sigma^{Es}$ (orange, thick), $\sigma^{El}$ (green), $\sigma^{Co}$ (purple), $\sigma^{fp}$ (dashed blue), and $\dot I_{A:B}$ (dot-dashed black), in the strong coupling regime, with $\omega_B/\omega_A = 0.6$, $\Delta/\omega_A = 0.4$, $\omega_A\beta_A = 1.1$, $\omega_A\beta_B = 0.3$, and $g/\omega_A =0.3$.
• Figure 3: Plots of the entropy production for several initial states of $A$ in the weak coupling regime with $\omega_B/\omega_A = 0.6$, $\Delta/\omega_A = 0.4$, $\omega_A\beta_A = 1.1$, $\omega_A\beta_B = 0.3$, and $g/\omega_A =0.03$. The shaded grey area corresponds to the intervals of time where the dynamics is P-divisible (see the text for details).
• Figure 4: (a) Plots of the entropy production $\sigma^{fp}$ for several initial states of $A$ and (b) plot of $\sigma^{fp}_\text{min}$, minimum of the entropy production over all initial states, in the weak coupling regime with $\omega_B/\omega_A = 0.6$, $\Delta/\omega_A = 0.4$, $\omega_A\beta_B = 0.3$, and $g/\omega_A =0.03$. The shaded grey areas correspond to the intervals of time where the dynamics is P-divisible.
• Figure 5: (a) Plot of $\sigma^{fp}_\text{min}$, minimum of the entropy production over all initial states, in the strong coupling regime $g/\omega_A =0.3$, near to resonance, $\omega_B/\omega_A = 0.99$, $\Delta/\omega_A = 0.01$, and cold bath $\omega_A\beta_B = 3$. (b) Zoom of plot (a) on a region where the equivalence between P-divisibility and the sign of $\sigma^{fp}_\text{min}$ fails. The blue dashed line corresponds to the plot of $\sigma_\text{map}$, the map entropy production defined in the main text in Eq. \ref{['eq:sigmamap']}. Note that the minimum over all states in the definition of $\sigma_\text{map}$ is obtained numerically by a discrete parameterization of the Bloch ball. However, for some instant of times, this minimum cannot be accessed numerically. Instead we perform an analytical analysis to obtain it. See Appendix \ref{['app:findingsigmamap']} for more details.

Theorems & Definitions (7)

• Theorem 1
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof