Non-classicality at equilibrium and efficient predictions under non-commuting charges

A quantum thermodynamic system can conserve non-commuting observables, but the consequences of this phenomenon on relaxation are still not fully understood. We investigate this problem by leveraging an observable-dependent approach to equilibration and thermalization in isolated quantum systems. We extend such approach to scenarios with non-commuting charges, and show that it can accurately estimate the equilibrium distribution of coarse observables without access to the energy eigenvalues and eigenvectors. Our predictions do not require weak coupling and are not restricted to local observables, thus providing an advantage over the non-Abelian thermal state. Within this approach, weak values and quasiprobability distributions emerge naturally and play a crucial role in characterizing the equilibrium distributions of observables. We show and numerically confirm that, due to charges' non-commutativity, these weak values can be anomalous even at equilibrium, which has been proven to be a proxy for non-classicality. Our work thus uncovers a novel connection between the relaxation of observables under non-commuting charges, weak values, and Kirkwood-Dirac quasiprobability distributions.