Entanglement of approximate quantum strategies in XOR games

TL;DR

This paper constructs specific XOR games demonstrating that achieving near-optimal bias requires a large amount of entanglement, significantly improving previous bounds on entanglement and noise tolerance in quantum self-tests.

Contribution

It introduces XOR games with minimal inputs that require substantial entanglement for near-optimal strategies, advancing understanding of entanglement complexity in quantum games.

Findings

Exponential improvement in input size and noise tolerance over previous self-tests.

Near-optimal strategies require entanglement scaling as \\varepsilon^{-1/5} in the constructed XOR games.

Tight bounds on entanglement needed for \\varepsilon$-optimal strategies in XOR games.

Abstract

We show that for any $\varepsilon>0$ there is an XOR game $G=G(\varepsilon)$ with $\Theta(\varepsilon^{-1/5})$ inputs for one player and $\Theta(\varepsilon^{-2/5})$ inputs for the other player such that $\Omega(\varepsilon^{-1/5})$ ebits are required for any strategy achieving bias that is at least a multiplicative factor $(1-\varepsilon)$ from optimal. This gives an exponential improvement in both the number of inputs or outputs and the noise tolerance of any previously-known self-test for highly entangled states. Up to the exponent $-1/5$ the scaling of our bound with $\varepsilon$ is tight: for any XOR game there is an $\varepsilon$-optimal strategy using $\lceil \varepsilon^{-1} \rceil$ ebits, irrespective of the number of questions in the game.