A quantum information-theoretic proof of the relation between horn's problem and the Littlewood-Richardson coefficients

Abstract

Horn's problem asks for the conditions on sets of integers μ, ν and λ that ensure the existence of Hermitian operators A, B and A∈+∈B with spectra μ, ν and λ, respectively. It has been shown that this problem is equivalent to deciding whether U λ ⊂ U μ ⊗ U ν for irreducible representations of with highest weights μ, ν and λ. In this paper we present a quantum information-theoretic proof of the relation between the two problems that is asymptotic in one direction. This result has previously been obtained by Klyachko using geometric invariant theory [1]. The work presented in this paper does not, however, touch upon the non-asymptotic equivalence between the two problems, a result that rests on the recently proven saturation conjecture for [2]. © 2008 Springer-Verlag Berlin Heidelberg.