Maximal lexicographic spectra and ranks for states with fixed uniform margins

Abstract

We find the spectrum in maximal lexicographic order for quantum states ρ AB ∈ H A ⊗ H B with margins ρ A = 1 n I n and ρ B = 1 m I m and discuss the construction of ρ AB . By nonzero rectangular Kronecker coefficients, we give counterexamples for Klyachko's conjecture which says that a quantum state with maximal lexicographical spectrum has minimal rank among all states with given margins. Moreover, we show that quantum states with the maximal lexicographical spectrum are extreme points.2010 Mathematics Subject Classification. Primary 20C30; Secondary 15A18.