An asymptotic property of factorizable completely positive maps and the Connes embedding problem

TL;DR

This paper links the Connes embedding problem to an asymptotic property of factorizable completely positive maps, proving factorizability for certain channels and exploring their asymptotic behaviors.

Contribution

It reformulates the Connes embedding problem via factorizable maps and proves factorizability for specific Holevo-Werner channels, advancing understanding of their asymptotic properties.

Findings

Holevo-Werner channels W_n^- are factorizable for all odd n ≠ 3

Factorizability of convex combinations of W_3^+ and W_3^- analyzed

Asymptotic properties of these channels discussed

Abstract

We establish a reformulation of the Connes embedding problem in terms of an asymptotic property of factorizable completely positive maps. We also prove that the Holevo-Werner channels W_n^- are factorizable, for all odd integers n different from 3. Furthermore, we investigate factorizability of convex combinations of W_3^+ and W_3^-, a family of channels studied by Mendl and Wolf, and discuss asymptotic properties for these channels.