Deep thermalization for mixed states
Phys. Rev. B 113, 174309 – Published 18 May, 2026
DOI: https://doi.org/10.1103/x397-3nk5
Abstract
Deep thermalization, where ensembles of pure states on a local subsystem, conditioned on measurement outcomes on its complement, approach universal maximum-entropy ensembles, represents a stronger form of ergodicity than conventional thermalization. We show that this framework fails dramatically for mixed initial states, evolved unitarily, even with infinitesimal initial mixedness. To address this, we introduce an alternate paradigm of deep thermalization for mixed states, fundamentally distinct from that for pure-state ensembles. In our formulation, the deep thermal ensemble arises by tracing out auxiliary degrees of freedom from a maximum-entropy ensemble defined on an augmented system, with the ensemble structure depending explicitly on the entropy of the initial state. We demonstrate that such ensembles emerge dynamically in generic, locally interacting chaotic systems. For the self-dual kicked Ising chain, which we show to be exactly solvable for a class of mixed initial states, we find exact emergence of the so-defined mixed-state deep thermal ensemble at finite times. Our results therefore lead to fundamental insights into how maximum-entropy principles and deep thermalization manifest themselves in unitary dynamics of states with finite entropy.
