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

• DOI: 10.1007/s00220-024-05122-3
• Publication Date: 2024-08-31T12:03:07Z
• Authors: Laura Mančinska; Jitendra Prakash; Christopher Schaf...
• Keywords: Quantum Physics; Mathematics - Operator Algebras; FOS: Mathematics; 81P40 (Primary) 81P40, 47C15 (Secondary); FOS: Physical sciences; 0102 computer and information sciences; Quantum Physics (quant-ph); Operator Algebras (math.OA); 01 natural sciences

38 pages<br/>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.<br/>