Random matrices with prescribed eigenvalues and expectation values for random quantum states

TL;DR

This paper investigates the statistical properties of random matrices with prescribed eigenvalues, showing that linear functionals of their entries tend to Gaussian distributions in large dimensions, with applications in quantum mechanics and matrix theory.

Contribution

It provides new bounds on distribution distances for linear functionals of such matrices and extends results to growing numbers of these functionals, connecting to quantum states and spectral theory.

Findings

Linear functionals are approximately Gaussian for large matrices.

Bounds on distribution distances are established.

Applications include quantum expectation values and spectral distributions.

Abstract

Given a collection $\{\lambda_1, \dots, \lambda_n\} $ of real numbers, there is a canonical probability distribution on the set of real symmetric or complex Hermitian matrices with eigenvalues $\lambda_1,\ldots,\lambda_n$. In this paper, we study various features of random matrices with this distribution. Our main results show that under mild conditions, when $n$ is large, linear functionals of the entries of such random matrices have approximately Gaussian joint distributions. The results take the form of upper bounds on distances between multivariate distributions, which allows us also to consider the case when the number of linear functionals grows with $n$. In the context of quantum mechanics, these results can be viewed as describing the joint probability distribution of the expectation values of a family of observables on a quantum system in a random mixed state. Other applications are given to spectral distributions of submatrices, the classical invariant ensembles, and to a probabilistic counterpart of the Schur--Horn theorem, relating eigenvalues and diagonal entries of Hermitian matrices.