Geometry of density matrix states

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 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.