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

## TL;DR

This paper analyzes the distribution of expectation values of arbitrary observables for random pure states in high-dimensional quantum systems, revealing a narrow Gaussian peak with non-Gaussian tails and explicit large deviation behavior in special cases.

## Contribution

It provides an analytical characterization of expectation value distributions for random states, including explicit formulas under Wigner's semicircle law, highlighting non-Gaussian tail behavior and phase transition-like features.

## Key findings

- Expectation values form a narrow Gaussian peak.
- Tails deviate significantly from Gaussian behavior.
- Explicit large deviation function for Wigner's law case.

## Abstract

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.