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

Magdalena Musat

TL;DR

This paper surveys the deep connections between the Connes Embedding Problem, Kirchberg's reformulations, Tsirelson's conjecture, and the theory of factorizable quantum channels. It highlights how correlation matrices of unitaries encode CEP obstructions and how factorizable maps can fail to be realizable with finite-dimensional ancillas, even as they arise from mixtures of unitaries. It provides a self-contained finite-dimensional proof of Tsirelson's theorem and presents new obstructions showing that some channels are not k-noisy for any k≥2, underscoring infinite-dimensional phenomena. The work also integrates recent breakthroughs (MIP* = RE) to explain why CEP has a negative resolution and discusses the resulting exotic ancilla behavior and non-closure phenomena in correlation-sets.

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).

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

TL;DR

This paper surveys the deep connections between the Connes Embedding Problem, Kirchberg's reformulations, Tsirelson's conjecture, and the theory of factorizable quantum channels. It highlights how correlation matrices of unitaries encode CEP obstructions and how factorizable maps can fail to be realizable with finite-dimensional ancillas, even as they arise from mixtures of unitaries. It provides a self-contained finite-dimensional proof of Tsirelson's theorem and presents new obstructions showing that some channels are not k-noisy for any k≥2, underscoring infinite-dimensional phenomena. The work also integrates recent breakthroughs (MIP* = RE) to explain why CEP has a negative resolution and discusses the resulting exotic ancilla behavior and non-closure phenomena in correlation-sets.

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 -noisy (admitting a factorization through a full matrix algebra).

Paper Structure

This paper contains 4 sections, 15 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. The Connes Embedding Problem and correlation matrices of unitaries
  3. Factorizable quantum channels and their ancillas
  4. The Connes-Kirchberg problem and Tsirelson's conjecture

Key Result

Proposition 2.1

Let ${\mathcal{N}}$ be a finite factor with separable predual. Then ${\mathcal{N}}$ admits a normal embedding into ${\mathcal{R}}^\omega$ if and only if for all integers $n,\ell \ge 0$, all unitaries $u_1, \dots, u_n$ in ${\mathcal{N}}$, and all ${\varepsilon} >0$, there exist $k \in {\mathbb N}$ an for all $r \ge 1$, $i_1, \dots, i_r \in \{1, \dots, n\}$, and $\nu_1, \dots, \nu_r \in {\mathbb Z}$

Theorems & Definitions (27)

  • Proposition 2.1: Connes
  • Theorem 2.2: Kirchberg
  • proof
  • Remark 2.3
  • Theorem 2.4: Musat--Rø rdam
  • Theorem 2.5: Mori
  • Definition 3.1
  • Remark 3.2: Stinespring representations
  • Example 3.3
  • Example 3.4: Mixtures of unitaries
  • ...and 17 more