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Transactions of the American Mathematical Society

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


Authors: Elizabeth S. Meckes and Mark W. Meckes
Journal: Trans. Amer. Math. Soc. 373 (2020), 5141-5170
MSC (2010): Primary 60B20; Secondary 81P15
DOI: https://doi.org/10.1090/tran/8074
Published electronically: April 28, 2020
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Abstract: Given a collection $ \underline {\lambda }=\{\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.


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Additional Information

Elizabeth S. Meckes
Affiliation: Department of Mathematics, Applied Mathematics, and Statistics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, Ohio 44106
Email: elizabeth.meckes@case.edu

Mark W. Meckes
Affiliation: Department of Mathematics, Applied Mathematics, and Statistics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, Ohio 44106
Email: mark.meckes@case.edu

DOI: https://doi.org/10.1090/tran/8074
Received by editor(s): July 30, 2019
Received by editor(s) in revised form: December 4, 2019
Published electronically: April 28, 2020
Additional Notes: This research was supported in part by grants from the U.S. National Science Foundation (DMS-1612589 to the first author) and the Simons Foundation (#315593 to the second author).
Article copyright: © Copyright 2020 American Mathematical Society