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 mu, nu and lambda that ensure the existence of Hermitian operators A, B and A+B with spectra mu, nu and lambda, respectively. It has been shown that this problem is equivalent to deciding whether the irreducible representation of GL(d) with highest weight lambda is contained in the tensor product of irreducible representations with highest weight mu and nu. 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. 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 GL(d).