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Cambridge University Press eBooks

Property (T) and residually finite groups

Gilles Pisier

Published Feb 10, 2020• 0 citations
Publisher

Abstract

This is a brief excursion into a topic that belongs to quantum information theory. We prove here that the Connes-Kirchberg problem is equivalent to a question raised by Tsirelson in connection with quantum mechanics. The question involves correlation matrices of various kinds that we discuss using operator valued measures in order to highlight the equivalence of the Tsirelson problem with the LLP of certain free product C*-algebras.

DOI
10.1017/9781108782081.018

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This is a brief excursion into a topic that belongs to quantum information theory. We prove here that the Connes-Kirchberg problem is equivalent to a question raised by Tsirelson in connection with quantum mechanics. The question involves correlation matrices of various kinds that we discuss using operator valued measures in order to highlight the equivalence of the Tsirelson problem with the LLP of certain free product C*-algebras.
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 kkk-noisy (admitting a factorization through a full matrix algebra).
En 1976, Connes demande si toute algèbre de von Neumann finie se plonge dans un ultraproduit d'algèbres de matrices. En 1980, Tsirelson demande si, dans la formulation mathématique de la mécanique quantique, autoriser des espaces de Hilbert de dimension infinie change fondamentalement le modèle. En 1993, Kirchberg conjecture que le produit tensoriel de deux copies de la C*-algèbre pleine du groupe libre de rang infini dénombrable peut être muni d'une unique norme de C*-algèbre. De manière surprenante et non triviale, ces trois problèmes sont en fait équivalents, c'est maintenant bien compris. Ces problèmes viennent d'être résolus, par la négative, avec des méthodes d'informatique : calculabilité, complexité, et informatique quantique. Je ferai de mon mieux pour raconter les grandes lignes de cette très longue preuve.
More precisely, we give a simple and very short proof of "the Connes embedding problem implies the synchronous Tsirelson conjecture" that relies on only two elementary ingredients: 1) the well-known description of synchronous correlations as traces on the algebra per player C∗(player)=C∗(inputs∣outputs)C^*(\mathrm{player})=C^*(\mathrm{inputs}|\mathrm{outputs})C∗(player)=C∗(inputs∣outputs) and 2) an elementary lifting result by Kim, Paulsen and Schafhauser. Moreover, this bypasses every of the deep results by Kirchberg as well as any other implicit reformulation as the microstates conjecture and thelike. Meanwhile, we also give a different construction of Connes' algebra Rω\mathcal{R}^ωRω appearing in the Connes embedding problem, which is more suitable for the purposes of quantum information theory and much easier to comprehend for the reader without any prior knowledge in operator algebras. Most importantly, however, we present this proof for the following reason: Since the recent refutation of the synchronous Tsirelson conjecture by MIP*=RE, there exists a nonlocal game which violates the synchronous Tsirelson conjecture, and by the proof of MIP*=RE even a synchronous such game. The approach however is based on contradiction with the undecidability of the Halting problem, and so remains implicit. As such the quest now has started to give an explicit example of a synchronous game violating the synchronous Tsirelson conjecture together with a direct argument for the failure, and the current article serves as a direct translation to the corresponding operator algebra and its tracial state violating the Connes embedding problem.
Connes’ Embedding Problem is a deep question on approximability of certain tracial von Neumann algebras by finite-dimensional matrix algebras. We survey the connections between operator algebras, quantum information and theoretical computer science that enabled the recent resolution of this problem. The resolution goes through an equivalent formulation, known as Tsirelson’s problem, in terms of separating convex sets whose definition is motivated by the study of nonlocality in quantum mechanics. We construct an explicit separating hyperplane using the theory of two-player games from complexity theory.