Probability density of quantum expectation values

TL;DR

This paper derives the exact probability distribution of quantum expectation values for pure states, providing insights into their statistical behavior and comparing with concentration bounds, with implications for quantum statistical mechanics.

Contribution

It presents the exact probability density function of quantum expectation values over pure states, using elementary methods and characteristic functions, for both degenerate and non-degenerate observables.

Findings

Exact probability density derived for quantum expectation values.

Comparison with Levy's lemma concentration bounds.

Insights into statistical mechanics on the energy shell.

Abstract

We consider the quantum expectation value \mathcal{A}=\<\psi|A|\psi\> of an observable A over the state |\psi\> . We derive the exact probability distribution of \mathcal{A} seen as a random variable when |\psi\> varies over the set of all pure states equipped with the Haar-induced measure. The probability density is obtained with elementary means by computing its characteristic function, both for non-degenerate and degenerate observables. To illustrate our results we compare the exact predictions for few concrete examples with the concentration bounds obtained using Levy's lemma. Finally we comment on the relevance of the central limit theorem and draw some results on an alternative statistical mechanics based on the uniform measure on the energy shell.