Bounds for Apparent Second-Law Violations in Quantum Trajectories

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 $\langle\Omega\rangle=\Sigma+\Sigma^\ast$. 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 $\Pi_\sigma=\Pr(\sigma<0)+\Pr(\sigma=0)/2$. When $\Pr(\sigma=0)=0$, the result reads $\Pr(\sigma<0)\ge[1-\langle\Omega\rangle/g(\langle\Omega\rangle)]/2$, 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.