Geometry of density matrix states

Abstract

We reconsider the geometry of pure and mixed states in a finite quantum system. The ranges of eigenvalues of the density matrices delimit a regular symplex (hypertetrahedron ${T}_{N}$) in any dimension $N$; the polytope isometry group is the symmetric group ${S}_{N+1}$, and splits ${T}_{N}$ in chambers, the orbits of the states under the projective group $\mathrm{PU}(N+1)$. 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.