Full expectation value statistics for randomly sampled pure states in high-dimensional quantum systems

Abstract

We explore how the expectation values $⟨\psi |A|\psi ⟩$ of a largely arbitrary observable $A$ are distributed when normalized vectors $|\psi ⟩$ are randomly sampled from a high dimensional Hilbert space. Our analytical results predict that the distribution exhibits a very narrow peak of approximately Gaussian shape, while the tails significantly deviate from a Gaussian behavior. In the important special case that the eigenvalues of $A$ satisfy Wigner's semicircle law, the expectation value distribution for asymptotically large dimensions is explicitly obtained in terms of a large deviation function, which exhibits two symmetric non-analyticities akin to critical points in thermodynamics.

References in corpus (4)

• The large deviation approach to statistical mechanics
• Typicality for Generalized Microcanonical Ensembles
• Typicality versus thermality: An analytic distinction
• Concentration of measure for quantum states with a fixed expectation value

Cited by in corpus (1)

• Why are macroscopic experiments reproducible? Imitating the behavior of an ensemble by single pure states