Improving the Cram\'er-Rao bound with the detailed fluctuation theorem

TL;DR

This paper derives an improved upper bound for the mean entropy production rate in non-equilibrium systems using the detailed fluctuation theorem, surpassing the traditional Cramér-Rao bound, with applications to heat exchange models.

Contribution

It introduces a novel bound for entropy production rate based on the DFT, providing a more accurate estimate than the CR bound in certain quantum heat exchange scenarios.

Findings

The new bound closely approximates the entropy production rate in weakly coupled bosonic systems.

The bound is exactly saturated in weakly coupled qubit systems.

The approach enhances understanding of entropy production in non-equilibrium thermodynamics.

Abstract

In some non-equilibrium systems, the distribution of entropy production $p(\Sigma)$ satisfies the detailed fluctuation theorem (DFT), $p(\Sigma)/p(-\Sigma)=\exp(\Sigma)$. When the distribution $p(\Sigma)$ shows time-dependency, the celebrated Cram\'{e}r-Rao (CR) bound asserts that the mean entropy production rate is upper bounded in terms of the variance of $\Sigma$ and the Fisher information with respect to time. In this letter, we employ the DFT to derive an upper bound for the mean entropy production rate that improves the CR bound. We show that this new bound serves as an accurate approximation for the entropy production rate in the heat exchange problem mediated by a weakly coupled bosonic mode. The bound is saturated for the same setup when mediated by a weakly coupled qubit.