A variant of Horn's problem and 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 \emph{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 $\rG\rU\rE(n)$ 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.

24 pages, LaTeX; a new result, i.e., Theorem 3.7, is added and several references are included

References in corpus (7)

• Average subentropy, coherence and entanglement of random mixed quantum states
• The Golden-Thompson inequality --- historical aspects and random matrix applications
• Eigenvalue statistics for the sum of two complex Wishart matrices
• The difference between two random mixed quantum states: exact and asymptotic spectral analysis
• Average coherence and its typicality for random mixed quantum states
• Co-rank 1 projections and the randomised Horn problem
• Volume of the set of locally diagonalizable bipartite states

Cited by in corpus (5)

• Large Deviation Principles via Spherical Integrals
• Harmonic analysis for rank-1 Randomised Horn Problems
• Spectral density of mixtures of random density matrices for qubits
• Co-rank 1 projections and the randomised Horn problem
• Duistermaat-Heckman measure and the mixture of quantum states