Maximal lexicographic spectra and ranks for states with fixed uniform margins

TL;DR

This paper characterizes the maximal lexicographic spectra of bipartite quantum states with fixed uniform marginals, provides counterexamples to a conjecture relating spectrum and rank, and shows these states are extreme points.

Contribution

It determines the spectrum in maximal lexicographic order for states with uniform margins and disproves Klyachko's conjecture using rectangular Kronecker coefficients.

Findings

Counterexamples to Klyachko's conjecture using Kronecker coefficients

States with maximal lexicographic spectrum are extreme points

Explicit construction of states with given uniform margins

Abstract

We find the spectrum in maximal lexicographic order for quantum states $\rho_{AB}\in\mathcal{H}_A\otimes \mathcal{H}_B$ with margins $\rho_A=\frac{1}{n}I_n$ and $\rho_B=\frac{1}{m}I_m$ and discuss the construction of $\rho_{AB}$. 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.