Possible consequences for physics of the negative resolution of Tsirelson's problem

TL;DR

This paper discusses the implications of a recent proof that resolves Tsirelson's problem negatively, showing that certain quantum correlations cannot be approximated by finite-dimensional systems, and explores four logical possibilities of this outcome.

Contribution

It identifies four fundamental logical possibilities resulting from the negative resolution of Tsirelson's problem and discusses open problems to determine which is correct.

Findings

Separation of $C_{qa}$ and $C_{qc}$ by a hyperplane is possible.

Correlations from commuting measurements cannot always be approximated by finite-dimensional tensor products.

Four logical scenarios arise from the negative resolution, each with different implications.

Abstract

In 2020, Ji et al. [arXiv:2001.04383 and Comm.~ACM 64}, 131 (2021)] provided a proof that the complexity classes $\text{MIP}^\ast$ and $\text{RE}$ are equivalent. This result implies a negative resolution of Tsirelson's problem, that is, $C_{qa}$ (the closure of the set of tensor product correlations) and $C_{qc}$ (the set of commuting correlations) can be separated by a hyperplane (that is, a Bell-like inequality). In particular, there are correlations produced by commuting measurements (a finite number of them and with a finite number of outcomes) on an infinite-dimensional quantum system which cannot be approximated by sequences of finite-dimensional tensor product correlations. Here, we point out that there are four logical possibilities of this result. Each possibility is interesting because it fundamentally challenges the nature of spacially separated systems in different ways. We list open problems for making progress for deciding which of the possibilities is correct.