# The set of quantum correlations is not closed

## TL;DR

This paper constructs a linear system non-local game demonstrating that the set of quantum correlations is not closed, providing new insights into quantum non-locality and the Tsirelson problem.

## Contribution

It introduces a novel non-local game that separates finite-dimensional and infinite-dimensional quantum strategies, and proves the undecidability of certain quantum strategy questions.

## Key findings

- The set of tensor-product quantum correlations is not closed.
- A new counterexample to the Tsirelson problem is provided.
- Decidability of perfect strategies for linear system games is proven to be undecidable.

## Abstract

We construct a linear system non-local game which can be played perfectly using a limit of finite-dimensional quantum strategies, but which cannot be played perfectly on any finite-dimensional Hilbert space, or even with any tensor-product strategy. In particular, this shows that the set of (tensor-product) quantum correlations is not closed. The constructed non-local game provides another counterexample to the "middle" Tsirelson problem, with a shorter proof than our previous paper (though at the loss of the universal embedding theorem). We also show that it is undecidable to determine if a linear system game can be played perfectly with a finite-dimensional strategy, or a limit of finite-dimensional quantum strategies.