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Relating the Bures Measure to the Cauchy Two-Matrix Model

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Abstract

The Bures metric is a natural choice in measuring the distance of density operators representing states in quantum mechanics. In the past few years a random matrix ensemble and the corresponding joint probability density function of its eigenvalues was identified. Moreover, a relation with the Cauchy two-matrix model was discovered but never thoroughly investigated, leaving open in particular the following question: How are the kernels of the Pfaffian point process of the Bures random matrix ensemble related to the ones of the determinantal point process of the Cauchy two-matrix model, and moreover, how can it be possible that a Pfaffian point process derives from a determinantal point process? We give a very explicit answer to this question. The aim of our work has a quite practical origin since the calculation of the level statistics of the Bures ensemble is highly mathematically involved while we know the statistics of the Cauchy two-matrix ensemble. Therefore, we solve the whole level statistics of a density operator drawn from the Bures prior.

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Correspondence to Mario Kieburg.

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Forrester, P.J., Kieburg, M. Relating the Bures Measure to the Cauchy Two-Matrix Model. Commun. Math. Phys. 342, 151–187 (2016). https://doi.org/10.1007/s00220-015-2435-4

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