Bounds for Apparent Second-Law Violations in Quantum Trajectories

Abstract

Negative stochastic entropy production is commonly called an apparent violation of the second law. In general quantum-trajectory dynamics, however, the physical entropy production $\sigma$ need not obey a forward detailed fluctuation theorem. A general arbitrary-coupling formulation identifies a dynamical-asymmetry term ${\sigma }^{\ast }$ that completes it into $\Omega =\sigma +{\sigma }^{\ast }$, whose mean is $⟨\Omega ⟩=\Sigma +{\Sigma }^{\ast }$. We prove that the likelihood-ratio sign is optimal among reversal-odd trajectory observables and use this fact to transfer an established sharp fluctuation-theorem floor to the tie-corrected physical sign statistic ${\Pi }_{\sigma }=\mathrm{Pr}(\sigma <0)+\mathrm{Pr}(\sigma =0)/2$. When $\mathrm{Pr}(\sigma =0)=0$, the result reads $\mathrm{Pr}(\sigma <0)\ge [1-⟨\Omega ⟩/g(⟨\Omega ⟩)]/2$, where $g$ is the inverse of $a\mapsto a\tanh (a/2)$. The physical integral fluctuation theorem simultaneously suppresses large negative events, producing a quantitative ``frequent but mild'' law, while the sign imbalance lower-bounds the hidden mean ${\Sigma }^{\ast }$. We formulate the measured-record protocol explicitly and illustrate and numerically audit the tie-corrected theorem in random finite-coupling collision models and a coherently driven qubit interacting with thermal ancillas.

8 pages, 2 figures. Supplemental Material: 10 pages, 1 figure