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

We explore how the expectation values $\langle\psi |A| \psi\rangle$ of a largely arbitrary observable $A$ are distributed when normalized vectors $|\psi\rangle$ 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.