Extreme Points and Factorizability for New Classes of Unital Quantum Channels

Abstract

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 $t|e_j⟩⟨e_j|\oplus V$ with $V\in M_{d-1}(\mathbf{C})$ unitary and $t\in (-1,1)$. 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 $t=-1/(d-1)$. 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 $M_3(\mathbf{C})\otimes M_3(\mathbf{C})$.

Improved discussion of the question on factorizability when d = 4 and t = -1/3 in new Section 4.6.2

References in corpus (2)

• An asymptotic property of factorizable completely positive maps and the Connes embedding problem
• Non-closure of quantum correlation matrices and factorizable channels that require infinite dimensional ancilla

Cited by in corpus (3)

• Group-covariant extreme and quasi-extreme channels
• Thermal operations from informational equilibrium
• On the rank of extremal marginal states