The Connes-Kirchberg Problem and infinite-dimensional phenomena in quantum information theory

TL;DR

This paper explores deep connections between the Connes Embedding Problem, quantum information theory, and quantum channels, presenting both an overview and new results on obstructions to certain channel factorizations.

Contribution

It provides new obstructions for factorizable quantum channels to be k-noisy, advancing understanding of infinite-dimensional phenomena in quantum information.

Findings

Identified obstructions for channels to be k-noisy

Connected the Connes Embedding Problem with quantum channel properties

Provided new insights into infinite-dimensional quantum phenomena

Abstract

We give an overview of results tying together a circle of problems connected to the Connes Embedding Problem, Kirchberg's reformulations thereof, Tsirelson's conjecture and its relation to quantum information theory, and a class of quantum channels, called factorizable, introduced by Anantharaman-Delaroche. While parts of the article are more expository, there are new results, including obstructions for channels to being $k$-noisy (admitting a factorization through a full matrix algebra).