Entropic trade-off relations in stochastic thermodynamics via replica Markov processes

Abstract

Traditional thermodynamic trade-off relations usually apply to quantities that depend linearly on probability distributions. In contrast, many important information-theoretic measures, such as entropies, are nonlinear and therefore difficult to analyze with existing frameworks. Motivated by replica methods in quantum information and spin-glass theory, we introduce replica Markov processes, i.e., Markovian dynamics of $K$ independent, identical copies, and derive trade-off relations that bound relative moments of replica observables in terms of the dynamical activity. By choosing appropriate replica observables, these inequalities translate into bounds on nonlinear quantities of the original single process. Focusing on entropic measures of uncertainty, we obtain upper bounds on both the time derivative and the value of the Tsallis entropy for general trajectory-observable distributions. Moreover, we derive upper bounds on the R\'enyi and Tsallis entropies, where the bounds involve the initial dynamical activity. Analogous bounds can be derived for the entropies of the state distribution. In particular, we provide upper bounds on both the time derivative and the value of the Tsallis entropy of state distributions. Moreover, we show that the R\'enyi and Tsallis entropies of the state distribution are bounded from above by terms involving the local escape rate from the initial node. We also illustrate how replicas can be used to study other nonlinear functionals such as extreme-value observables. Finally, we extend the construction to continuously monitored open quantum systems, where the bounds are expressed in terms of the quantum dynamical activity. These results provide entropic counterparts to activity-based uncertainty relations and establish a general method for constraining nonlinear information-theoretic quantities in stochastic and quantum thermodynamics.