Maximal lexicographic spectra and ranks for states with fixed 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.

12pages, some proofs have been shortened and new references were added

References in corpus (5)

• The Spectra of Density Operators and the Kronecker Coefficients of the Symmetric Group
• Quantum marginal problem and representations of the symmetric group
• Quantum state transformations and the Schubert calculus
• Nonvanishing of Kronecker coefficients for rectangular shapes
• On (almost) extreme components in Kronecker products of characters of the symmetric groups