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

Abstract

Let p be an odd prime and let r be the smallest generator of the multiplicative group Z∗p. We show that there exists a correlation of size Θ(r2) that self-tests a maximally entangled state of local dimension p − 1. The construction of the correlation uses the embedding procedure proposed by Slofstra (Forum of Mathematics, Pi. (2019)). Since there are infinitely many prime numbers whose smallest multiplicative generator is in the set {2, 3, 5} (D.R. Heath-Brown The Quarterly Journal of Mathematics (1986) and M. Murty The Mathematical Intelligencer (1988)), our result implies that constant-sized correlations are sufficient for self-testing of maximally entangled states with unbounded local dimension.