Practical Parallel Self-testing of Bell States via Magic Rectangles.

Self-testing is a method to verify that one has a particular quantum state from purely classical statistics. For practical applications, such as device-independent delegated verifiable quantum computation, it is crucial that one self-tests multiple Bell states in parallel while keeping the quantum capabilities required of one side to a minimum. In this work we use the $3 \times n$ magic rectangle games (generalisations of the magic square game) to obtain a self-test for $n$ Bell states where the one side needs only to measure single-qubit Pauli observables. The protocol requires small input size (constant for Alice and $O(\log n)$ bits for Bob) and is robust with robustness $O(n^{5/2} \sqrt{\varepsilon})$, where $\varepsilon$ is the closeness of the observed correlations to the ideal. To achieve the desired self-test we introduce a one-side-local quantum strategy for the magic square game that wins with certainty, generalise this strategy to the family of $3 \times n$ magic rectangle games, and supplement these nonlocal games with extra check rounds (of single and pairs of observables).