# A two-player dimension witness based on embezzlement, and an elementary proof of the non-closure of the set of quantum correlations

## TL;DR

This paper introduces a two-player non-local game demonstrating that near-optimal strategies require exponentially large entangled states, and offers an elementary proof of the non-closure of quantum correlation sets based on embezzlement.

## Contribution

It presents a simplified two-player game inspired by a three-player version, reducing complexity and providing an elementary proof of a fundamental quantum information concept.

## Key findings

- Near-optimal strategies require exponentially large entangled states.
- The game reduces the number of questions and answers compared to previous models.
- Provides an elementary proof of the non-closure of quantum correlations.

## Abstract

We describe a two-player non-local game, with a fixed small number of questions and answers, such that an $\epsilon$-close to optimal strategy requires an entangled state of dimension $2^{\Omega(\epsilon^{-1/8})}$. Our non-local game is inspired by the three-player non-local game of Ji, Leung and Vidick [arXiv:1802.04926]. It reduces the number of players from three to two, as well as the question and answer set sizes. Moreover, it provides an (arguably) elementary proof of the non-closure of the set of quantum correlations, based on embezzlement and self-testing. In contrast, previous proofs involved representation theoretic machinery for finitely-presented groups and $C^*$-algebras.