Geometry of generalized depolarizing channels

A generalized depolarizing channel acts on an N-dimensional quantum system to compress the “Bloch ball” in N²−1 directions; it has a corresponding compression vector. We investigate the geometry of these compression vectors and prove a conjecture of Dixit and Sudarshan [ Phys. Rev. A 78 032308 (2008)], namely, that when N=2^d (i.e., the system consists of d qubits), and we work in the Pauli basis then the set of all compression vectors forms a simplex. We extend this result by investigating the geometry in other bases; in particular we find precisely when the set of all compression vectors forms a simplex.