Geometry of density sates

TL;DR

This paper explores the geometric structure of quantum states in finite systems, revealing how eigenvalue ranges form a regular simplex and how this relates to state purity, symmetry, and entropy.

Contribution

It provides a detailed geometric characterization of quantum states using simplices and explores the implications for entropy and invariants in finite quantum systems.

Findings

Eigenvalues of density matrices form a regular simplex in any dimension.

The simplex's symmetry group is the symmetric group SN+1.

Entropy surfaces can be visualized within this geometric framework.

Abstract

We reconsider the geometry of pure and mixed states in a finite quantum system. The rangesof eigenvalues of the density matrices delimit a regular simplex (Hypertetrahedron TN) in any dimension N; the polytope isometry group is the symmetric group SN+1, and splits TN in chambers, the orbits of the states under the projective group PU(N + 1). The type of states correlates with the vertices, edges, faces, etc. of the polytope, with the vertices making up a base of orthogonal pure states. The entropy function as a measure of the purity of these states is also easily calculable; we draw and consider some isentropic surfaces. The Casimir invariants acquire then also a more transparent interpretation.