Maximal lexicographic spectra and ranks for states with fixed uniform margins

Abstract

We find the spectrum in maximal lexicographic order for quantum states ρ A B ∈ H A ⊗ H B \rho _{AB}\in \mathcal {H}_A\otimes \mathcal {H}_B with margins ρ A = 1 n I n \rho _A=\frac {1}{n}I_n and ρ B = 1 m I m \rho _B=\frac {1}{m}I_m and discuss the construction of ρ A B \rho _{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.