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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
93/2019

A variant of Horn's problem and the derivative principle

Lin Zhang and Hua Xiang

Abstract

Identifying the spectrum of the sum of two given Hermitian matrices with fixed eigenvalues is the famous Horn's problem. In this note, we investigate a variant of Horn's problem, i.e., we identify the probability density function (abbr. pdf) of the diagonals of the sum of two random Hermitian matrices with given spectra. We then use it to re-derive the pdf of the eigenvalues of the sum of two random Hermitian matrices with given eigenvalues via derivative principle, a powerful tool used to get the exact probability distribution by reducing to the corresponding distribution of diagonal entries.We can recover Jean-Bernard Zuber's recent results on the pdf of the eigenvalues of two random Hermitian matrices with given eigenvalues. Moreover, as an illustration, we derive the analytical expressions of eigenvalues of the sum of two random Hermitian matrices from GUE(n) or Wishart ensemble by derivative principle, respectively.We also investigate the statistics of exponential of random matrices and connect them with Golden-Thompson inequality, and partly answer a question proposed by Forrester. Some potential applications in quantum information theory, such as uniform average quantum Jensen-Shannon divergence and average coherence of uniform mixture of two orbits, are discussed.

Received:
26.09.2019
Published:
04.12.2019
MSC Codes:
22E70, 81Q10, 46L30
Keywords:
Horn’s problem, derivative principle, probability density function