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Self-testing protocols enable the certification of quantum devices without demanding full knowledge about their inner workings. A typical approach in designing such protocols is based on observing nonlocal correlations which exhibit maximum violation in a Bell test. We show that in the Bell experiment known as Hardy's test of nonlocality, not only does the maximally nonlocal correlation self-test a quantum state, rather a nonmaximal nonlocal behavior can serve the same purpose. We, in fact, completely characterize all such behaviors leading to a self-test of every pure two-qubit entangled state except for the maximally entangled ones. Apart from presenting an alternative self-testing protocol, our method provides a powerful tool towards characterizing the complex boundary of the set of quantum correlations.
Rai et al. (Fri,) studied this question.