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Given a collection $\{\lambda_1, \dots, \lambda_n\} $ of real numbers, thereis a canonical probability distribution on the set of real symmetric or complexHermitian matrices with eigenvalues $\lambda_1,\ldots,\lambda_n$. In thispaper, we study various features of random matrices with this distribution. Ourmain results show that under mild conditions, when $n$ is large, linearfunctionals of the entries of such random matrices have approximately Gaussianjoint distributions. The results take the form of upper bounds on distancesbetween multivariate distributions, which allows us also to consider the casewhen the number of linear functionals grows with $n$. In the context of quantummechanics, these results can be viewed as describing the joint probabilitydistribution of the expectation values of a family of observables on a quantumsystem in a random mixed state. Other applications are given to spectraldistributions of submatrices, the classical invariant ensembles, and to aprobabilistic counterpart of the Schur--Horn theorem, relating eigenvalues anddiagonal entries of Hermitian matrices.

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