Connes implies Tsirelson: a simple proof

TL;DR

This paper provides a simple, elementary proof that Connes' embedding problem implies the synchronous Tsirelson conjecture, avoiding deep operator algebra results and offering insights relevant to quantum information theory.

Contribution

It offers a straightforward proof linking Connes' embedding problem to the synchronous Tsirelson conjecture using elementary tools and constructs a more accessible version of Connes' algebra for quantum information applications.

Findings

A simple proof connecting Connes' problem to Tsirelson conjecture

A new construction of Connes' algebra suitable for quantum info

Implications for nonlocal games and quantum correlations

Abstract

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 $C^*(\mathrm{player})=C^*(\mathrm{inputs}|\mathrm{outputs})$ 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 $\mathcal{R}^\omega$ 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.