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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 $p(\Sigma)$ showstime-dependency, the celebrated Cram\'{e}r-Rao (CR) bound asserts that the meanentropy 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 theDFT to derive an upper bound for the mean entropy production rate that improvesthe CR bound. We show that this new bound serves as an accurate approximationfor the entropy production rate in the heat exchange problem mediated by aweakly coupled bosonic mode. The bound is saturated for the same setup whenmediated by a weakly coupled qubit.

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