Constant-sized correlations are sufficient to robustly self-test maximally entangled states with unbounded dimension

TL;DR

This paper demonstrates that constant-sized correlations can robustly self-test maximally entangled states of unbounded dimension for infinitely many prime numbers, using an embedding procedure and properties of prime generators.

Contribution

It proves that constant-sized correlations suffice for robust self-testing of high-dimensional maximally entangled states, extending previous results to unbounded dimensions.

Findings

Constant-sized correlations can self-test maximally entangled states of unbounded dimension.

The construction applies to infinitely many primes with small multiplicative generators.

The embedding procedure enables robust self-testing in high-dimensional quantum systems.

Abstract

We show that for any prime odd integer $d$, there exists a correlation of size $\Theta(r)$ that can robustly self-test a maximally entangled state of dimension $4d-4$, where $r$ is the smallest multiplicative generator of $\mathbb{Z}_d^\ast$. The construction of the correlation uses the embedding procedure proposed by Slofstra (Forum of Mathematics, Pi. Vol. $7$, ($2019$)). Since there are infinitely many prime numbers whose smallest multiplicative generator is at most $5$ (M. Murty The Mathematical Intelligencer $10.4$ ($1988$)), our result implies that constant-sized correlations are sufficient for robust self-testing of maximally entangled states with unbounded local dimension.