The Bregman operator divergence is introduced for density matrices by differentiation of the matrix-valued function x ↦ x log x. This quantity is compared with the relative operator entropy of Fujii and Kamei. It turns out that the trace is the usual
Umegaki’s relative entropy which is the only intersection of the classes of quasi-entropies and Bregman divergences.
The Pauli channel acting on 2 × 2 matrices is generalized to an n-level quantum system. When the full matrix algebra Mn is decomposed into pairwise complementary subalgebras, then trace-preserving linear mappings Mn → Mn are constructed such that the restriction to the subalgebras are depolarizing channels. The result is the necessary and sufficient
condition of complete positivity. The main examples appear on bipartite systems.