Quantum channels can be mathematically represented as completely positive trace-preserving maps that act on a density matrix. A general quantum channel can be written as a convex sum of `extremal' channels. We show that for an $N$-level system, the extremal channel can be characterized in terms of $N^2$-$N$ real parameters coupled with rotations. We give a representation for $N$= 2, 3, 4.

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 SN+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.

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 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.

Depolarizing maps acting on an N-dimensional system are completely positive maps resulting into compression of the Bloch "ball" along N-2-1 polarization directions. In the qubit case these maps are a convex sum of four extremal maps and form a simplex in the space of compression coefficients along the three polarization directions. We calculate the compression domain for three- and four-level systems. For a three- level system the region has curved surfaces, but it is a simplex for a four-level system. We conjecture that it is a simplex in the case of 2(n)-level systems.