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

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 ${\lambda }_1,\dots ,{\lambda }_n$. 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.

v5: Minor revisions based on referee's comments; version to appear in Trans. Amer. Math. Soc

References in corpus (5)

• Random bosonic states for robust quantum metrology
• Asymptotics of random density matrices
• On the mean density of complex eigenvalues for an ensemble of random matrices with prescribed singular values
• Exponential bounds for the support convergence in the Single Ring Theorem
• Spectral statistics of the uni-modular ensemble

Cited by in corpus (1)

• Fluctuations of the spectrum in rotationally invariant random matrix ensembles