Nature is stingy: Universality of Scrooge ensembles in quantum many-body systems (2601.00266v1)

Abstract: Recent advances in quantum simulators allow direct experimental access to the ensemble of pure states generated by measuring part of an isolated quantum many-body system. These projected ensembles encode fine-grained information beyond thermal expectation values and provide a new window into quantum thermalization. In chaotic dynamics, projected ensembles exhibit universal statistics, a phenomenon known as deep thermalization. While infinite-temperature systems generate Haar-random ensembles, realistic physical constraints such as finite temperature or conservation laws require a more general framework. It has been proposed that deep thermalization is governed in general by the emergence of Scrooge ensembles, maximally entropic distributions of pure states consistent with the underlying constraints. Here we provide rigorous arguments supporting this proposal. To characterize this universal behavior, we invoke Scrooge $k$-designs, which approximate Scrooge ensembles, and identify three physically distinct mechanisms for their emergence. First, global Scrooge designs can arise from long-time chaotic unitary dynamics alone, without the need for measurements. Second, if the global state is highly scrambled, a local Scrooge design is induced when the complementary subsystem is measured. Third, a local Scrooge ensemble arises from an arbitrary entangled state when the complementary system is measured in a highly scrambled basis. Numerical simulations across a range of many-body systems identify coherence, entanglement, non-stabilizerness, and information scrambling as essential resources for the emergence of Scrooge-like behavior. Taken together, our results establish a unified theoretical framework for the emergence of maximally entropic, information-stingy randomness in quantum many-body systems.