Unital completely positive maps and their operator systems

TL;DR

This paper studies unital operator systems and their associated $C^*$-algebras, establishing conditions for isomorphisms and characterizing extreme points of unital completely positive maps.

Contribution

It proves the uniqueness of extending unital complete order isomorphisms to $C^*$-isomorphisms based on complete ranks and characterizes extreme points of unital CP maps up to co-cycle conjugacy.

Findings

Unique extension of isomorphisms when complete ranks match

Operator systems determined by the span of $v_iv_j^*$ for CP maps

Characterization of extreme points of unital CP maps

Abstract

A vector subspace $\cls$ of $\IM_n(\IC)$ is called unital operator system if $x \in \cls$ if and only if $x^* \in \cls$ and the identity operator $I_n \in \cls$, where $n$ is any fixed positive integer. Let $C^*(\cls)$ be the $C^*$ sub-algebra of $\IM_n(\IC)$ generated by the operator system $\cls$. We prove that a unital complete order isomorphism $\cli:\cls \raro \cls'$ between two such operator systems $\cls$ and $\cls'$ of $\IM_n(\IC)$ has a unique extension to a $C^*$-isomorphism $\cli:C^*(\cls) \raro C^*(\cls')$ if and only if $\cls$ and $\cls'$ are having equal set of complete ranks. The operator system $\cls = \mbox{span}\{v_iv_j^*:1 \le i,j \le d \}$ is uniquely determined for a unital completely positive map $\tau(x)=\sum_{1 \le k \le d} v_kxv_k^*$ of index $d \ge 1$. As an application of our main result, we explore this correspondence and characterize up to co-cycle conjugacy all extreme points in the convex set of unital completely positive maps on $\IM_n(\IC)$. Using the main result, we also characterize up to co-cycle conjugacy all extreme elements in the convex set of normalized trace preserving unital completely positive maps on $\IM_n(\IC)$.