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Analytic robustness bound for self-testing of the singlet with two binary measurements

Self-testing refers to a way of uniquely identifying an uncharacterized quantum device based solely on certain extremal quantum correlations. It is known that in the ideal case of a two-qubit singlet all the extremal points that can be obtained by measuring the singlet form the whole criteria for self-testing the singlet and the associated measurements. To make these self-testing criteria more suitable for practical applications, relevant robustness bounds are necessary to discuss. However, the previous robustness analyses for these criteria were obtained by the swap method, which may not be guaranteed to be tight. In this paper, we construct a general extraction map applying to all self-testing criteria of the singlet state with two binary measurements for each party and derive analytic robustness bounds for a self-testing singlet in different regions of the boundary of the extremal quantum correlation set. The comparison shows that our robustness bounds are better than the known results and close to optimal for these self-testing criteria of a singlet state with two binary measurements.