Extreme Points and Factorizability for New Classes of Unital Quantum Channels

We introduce and study two new classes of unital quantum channels. The first class describes a 2-parameter family of channels given by completely positive (CP) maps $M_3(\mathbf{C})\mapsto M_3(\mathbf{C})$ which are both unital and trace-preserving. Almost every member of this family is factorizable and extreme in the set of CP maps which are both unital and trace-preserving, but is not extreme in either the set of unital CP maps or the set of trace-preserving CP maps. We also study a large class of maps which generalize the Werner-Holevo channel for $d=3$ in the sense that they are defined in terms of partial isometries of rank $d-1$. Moreover, we extend this to maps whose Kraus operators have the form $t\mathrm{\mid }{e}_j⟩⟨e_j\mathrm{\mid }\oplus V$ with $V \in M_{d-1} ({\bf C}) $unitaryand$t \in (-1,1)$. We show that almost every map in this class is extreme in both the set of unital CP maps and the set of trace-preserving CP maps. We analyze in detail a particularly interesting subclass which is extreme unless $t=-1\mathrm{/}(d-1)$. For $d=3$, this includes a pair of channels which have a dual factorization in the sense that they can be obtained by taking the partial trace over different subspaces after using the same unitary conjugation in $M_3(\mathbf{C})\otimes M_3(\mathbf{C})$.