Random matrices with prescribed eigenvalues and expectation values for random quantum states
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- by Elizabeth S. Meckes and Mark W. Meckes PDF
- Trans. Amer. Math. Soc. 373 (2020), 5141-5170 Request permission
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.References
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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
- MR Author ID: 754850
- 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
- MR Author ID: 729101
- Email: mark.meckes@case.edu
- 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).
- © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 5141-5170
- MSC (2010): Primary 60B20; Secondary 81P15
- DOI: https://doi.org/10.1090/tran/8074
- MathSciNet review: 4127873