A Maximum Entropy Principle in Deep Thermalization and in Hilbert-Space Ergodicity
arxiv(2024)
摘要
We report universal statistical properties displayed by ensembles of pure
states that naturally emerge in quantum many-body systems. Specifically, two
classes of state ensembles are considered: those formed by i) the temporal
trajectory of a quantum state under unitary evolution or ii) the quantum states
of small subsystems obtained by partial, local projective measurements
performed on their complements. These cases respectively exemplify the
phenomena of "Hilbert-space ergodicity" and "deep thermalization." In both
cases, the resultant ensembles are defined by a simple principle: the
distributions of pure states have maximum entropy, subject to constraints such
as energy conservation, and effective constraints imposed by thermalization. We
present and numerically verify quantifiable signatures of this principle by
deriving explicit formulae for all statistical moments of the ensembles;
proving the necessary and sufficient conditions for such universality under
widely-accepted assumptions; and describing their measurable consequences in
experiments. We further discuss information-theoretic implications of the
universality: our ensembles have maximal information content while being
maximally difficult to interrogate, establishing that generic quantum state
ensembles that occur in nature hide (scramble) information as strongly as
possible. Our results generalize the notions of Hilbert-space ergodicity to
time-independent Hamiltonian dynamics and deep thermalization from infinite to
finite effective temperature. Our work presents new perspectives to
characterize and understand universal behaviors of quantum dynamics using
statistical and information theoretic tools.
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