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 (C) → M 3 (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 | ⊕ V with V ∈ M d−1 (C) unitary and t ∈ (−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 (C) ⊗ M 3 (C).

Group-covariant extreme and quasiextreme channels

“…Only special types of channels have been fully characterized to date: a set of qubit channels including all extreme points [9][10][11][12] and some extreme points for the set of unital channels [13,14], which are quantum counterparts of classical bistochastic processes. Studying unital channels is useful for investigating similarities between classical and quantum processes, such as establishing a quantum version of Birkhoff's Theorem [13] and proving additivity or superadditivity of quantities relevant to the communication capacity of channels [15][16][17].…”