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

Quantum 6, 614 (2022).

https://doi.org/10.22331/q-2022-01-03-614

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