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

Abstract

We show that for any prime odd integer d, there exists a correlation of size Θ(r) that can robustly self-test a maximally entangled state of dimension 4d − 4, where r is the smallest multiplicative generator of Z × d . 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.

Constant-sized robust self-tests for states and measurements of unbounded dimension

“…When compared to [Fu19], the strength of our proof is the simplicity and the small correlation size while the weakness is the non-constructive ( , ) dependence in the robustness proof.…”

“…In their work [Fu19], the author establishes certain algebraic relationships between the measurements needed to induce the considered constant-sized correlations but they do not present a self-test of measurements according to the standard definition [ŠB20]. Therefore, our work is the Nevertheless, we believe that a continuation of the arguments presented in [Fu19] should lead to constant-sized self-tests of measurements. first one to show that measurements of arbitrarily large dimension can be self-tested from constantsized correlations.…”

“…For applications, it would be efficient to have small-sized correlations that robustly self-test states with large dimensions. The only family of constant-sized correlations that self-test states of arbitrarily large dimension [Fu19] are constructed by building upon Slofstra's group embedding procedure [Slo19] for linear binary constraint system games. Specifically [Fu19] shows that for each ∈ , where is an infinite subset of the primes given by = { : is an odd prime, and the smallest generator of the group Z × is 2, 3 or 5}, the maximally entangled state 4( −1) can be robustly self-tested from correlations with over 100 questions per party.…”

“…The only family of constant-sized correlations that self-test states of arbitrarily large dimension [Fu19] are constructed by building upon Slofstra's group embedding procedure [Slo19] for linear binary constraint system games. Specifically [Fu19] shows that for each ∈ , where is an infinite subset of the primes given by = { : is an odd prime, and the smallest generator of the group Z × is 2, 3 or 5}, the maximally entangled state 4( −1) can be robustly self-tested from correlations with over 100 questions per party. We obtain the following corollary which complements the result in [Fu19], but with correlations of considerably smaller size:…”

“…Specifically [Fu19] shows that for each ∈ , where is an infinite subset of the primes given by = { : is an odd prime, and the smallest generator of the group Z × is 2, 3 or 5}, the maximally entangled state 4( −1) can be robustly self-tested from correlations with over 100 questions per party. We obtain the following corollary which complements the result in [Fu19], but with correlations of considerably smaller size:…”