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Random Matrices and Quantum Information Theory

PARRAUD, Félix, 京都大学 DOI:10.14989/doctor.k23449

2021.09.24

概要

This PhD lies at the intersection of Random Matrix Theory and Free Probability Theory. The connection between those two fields dates back to the early nineties with the work of Voiculescu who created the Theory of free probability. A probability theory for noncommutative variables where the notion of freeness replaces the one of independence in classical probability theory. This connection has yielded numerous results. In the first paper written during this P.h.D., we got, through new methods, an estimate on the non renormalized trace of some smooth functions in GUE matrices which in turns gives us a different proof of the main result of Haagerup and Thorbjørnsen. That is, the convergence of the norm of any polynomials in independent GUE random matrices. Our result also allows to consider polynomials in GUE matrices, deterministic matrices and tensor matrices. In the second paper written during this P.h.D., we prove similar results but for Haar unitary matrices instead of GUE matrices by adapting our method to the unitary case. Although some of those results were already proved by Collins and Male, this new method has the advantage of giving us quantitative estimates. In the third paper written during this P.h.D., we refine the method used in the first paper to prove a finite Taylor expansion around the dimension N of the non renormalized trace of some smooth functions in GUE matrices of size N. This also gives several interesting corollary on the local spectrum of polynomials in independent GUE matrices. In the fourth paper written during this P.h.D., we use the estimates obtained in the second paper to get measure concentration estimates on a random set. This set is defined as the output of a very specific quantum channel which is related to an important problem in Quantum Information Theory. The concentration estimates that we obtained give us explicit parameters of the size of the solutions. Finally in the fifth paper written during this P.h.D., we expand Voiculescu's result of convergence in law of the empirical measure of non-commutative polynomials in independent GUE matrices to non-commutative rational function.

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