Stratificational and antipodean properties of boundary states for N x N density matrices

TL;DR

This paper explores the geometric structure of the space of N x N density matrices, revealing stratification and antipodean properties of boundary states based on eigenvalue configurations.

Contribution

It introduces new geometric insights into the boundary states of density matrices, including stratification and antipodean symmetry related to eigenvalue distributions.

Findings

Boundary states with p zero eigenvalues lie on or outside specific spheres.

Existence of antipodean boundary states with complementary eigenvalue configurations.

Explicit radii formulas for spheres containing boundary states.

Abstract

We investigate the space of N x N dimensional density matrices. We show that there exist strata such that boundary states \rho_{p} with p zero eigenvalues lie on or outside the spheres with radii r_{p}=\sqrt{p/N(N-p)}. Moreover, we show that if in a certain direction there is a boundary state with q=N-p equal eigenvalues, then in the opposite (antipodean) direction exists a boundary state with p=N-q equal eigenvalues.