The dynamics of systems consisting of many particles are very complicated and practically impossible to predict. For such systems, statistical physics has proven extremely useful, making highly accurate predictions for both quantum and classical systems. The central assumption behind statistical physics is the maximum entropy principle: A generic system reaches a state with maximum entropy, up to its conservation laws such as of total energy or particle number. In this work, we investigate to what extent the maximum entropy principle holds in quantum many-body systems. Surprisingly, we find that it holds much more broadly and strongly in quantum systems than previously considered.

We develop a maximum entropy principle that provides a simple description of two aspects of quantum many-body systems. The first is Hilbert-space ergodicity, or how quantum dynamics uniformly fills its available space, respecting constraints such as energy conservation. The second is deep thermalization, a generalization of conventional quantum thermalization where, instead of ignoring the thermal bath, we perform measurements on it. Understanding both phenomena requires refining our definition of entropy, in which we may have access to a larger amount of information than is typically considered.

Our maximum entropy principle concretely relates to how natural quantum dynamics hides information, a notion that has been developed in recent years. It predicts a specific form of deep thermalization away from infinite temperature, which had previously not been understood. We anticipate that the simplicity of our principle will lend predictive power for complex quantum many-body systems in different settings, and that our principle will be relevant for both understanding the basic science of thermalization and designing useful quantum information tasks with many-body dynamics.