A variant of Horn's problem and the derivative principle

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 the derivative principle, a powerful tool used to get the exact probability distribution by reducing to the corresponding distribution of diagonal entries. We can also recover Jean-Bernard Zuber's recent results. 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 the derivative principle, respectively. We also investigate the statistics of exponential of random matrices and connect them with Golden-Thompson inequality, and partially 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.