Thermodynamic entropic uncertainty relation

Thermodynamic uncertainty relations reveal a fundamental trade-off between the precision of a trajectory observable and entropy production, where the uncertainty of the observable is quantified by its variance. In information theory, Shannon entropy is a common measure of uncertainty. However, a clear quantitative relationship between the Shannon entropy of an observable and the entropy production in stochastic thermodynamics remains to be established. In this study, we show that an uncertainty relation can be formulated in terms of the Shannon entropy of an observable and the entropy production. We introduce symmetry entropy, an entropy measure that quantifies the symmetry of the observable distribution, and demonstrate that a greater asymmetry in the observable distribution requires higher entropy production. Specifically, we establish that the sum of the entropy production and the symmetry entropy cannot be less than ln2. As a corollary, we also prove that the sum of the entropy production and the Shannon entropy of the observable is no less than ln2. As an application, we demonstrate our relation in the diffusion decision model, revealing a fundamental trade-off between decision accuracy and entropy production in stochastic decision-making processes.