Extreme Points and Factorizability for New Classes of Unital Quantum Channels

TL;DR

This paper introduces new classes of unital quantum channels, analyzing their extremality and factorizability properties, with detailed results for specific parameter ranges and generalizations of Werner-Holevo channels.

Contribution

It defines and studies two novel classes of unital quantum channels, characterizing their extremality, factorizability, and generalizations of Werner-Holevo channels in the 3-dimensional case.

Findings

Almost all channels in the first class are factorizable and extreme, but not in individual sets.

Most channels in the second class are extreme in both unital and trace-preserving sets.

A specific subclass is extreme except at a particular parameter value, with dual factorization properties for d=3.

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 $M_3({\bf C}) \mapsto M_3({\bf C})$ 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 \rangle \langle e_j | \oplus V $ with $V \in M_{d-1} ({\bf 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 $M_3({\bf C}) \otimes M_3({\bf C})$.