Constant-Sized Robust Self-Tests for States and Measurements of Unbounded Dimension

We consider correlations, p_n,x, arising from measuring a maximally entangled state using n measurements with two outcomes each, constructed from n projections that add up to xI. We show that the correlations p_n,x robustly self-test the underlying states and measurements. To achieve this, we lift the group-theoretic Gowers–Hatami based approach for proving robust self-tests to a more natural algebraic framework. A key step is to obtain an analogue of the Gowers–Hatami theorem allowing to perturb an “approximate” representation of the relevant algebra to an exact one. For n=4, the correlations p_n,x self-test the maximally entangled state of every odd dimension as well as 2-outcome projective measurements of arbitrarily high rank. The only other family of constant-sized self-tests for strategies of unbounded dimension is due to Fu (QIP 2020) who presents such self-tests for an infinite family of maximally entangled states with even local dimension. Therefore, we are the first to exhibit a constant-sized self-test for measurements of unbounded dimension as well as all maximally entangled states with odd local dimension.