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

We explore how the expectation values 〈ψ|A|ψ〉 of a largely arbitrary observable A are distributed when normalized vectors |ψ〉 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 nonanalyticities akin to critical points in thermodynamics.