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

We show that for any prime odd integer $d$, there exists a correlation of size $\mathrm{\Theta }(r)$ 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.