Weak form of self-testing

The concept of self-testing (or rigidity) refers to the fact that for certain Bell inequalities the maximal violation can be achieved in an essentially unique manner. In this work we present a family of Bell inequalities which are maximally violated by multiple inequivalent quantum realizations. We completely characterize the quantum realizations achieving the maximal violation and we show that each of them requires a maximally entangled state of two qubits. This implies the existence of a new, weak form of self-testing in which the maximal violation allows us to identify the state, but does not fully determine the measurements. From the geometric point of view the set of probability points that saturate the quantum bound is a line segment. We then focus on a particular member of the family and show that the self-testing statement is robust, i.e., that observing a nonmaximal violation allows us to make a quantitative statement about the unknown state. To achieve this we present a new construction of extraction channels and analyze their performance. For completeness we provide two independent approaches: analytical and numerical. The noise robustness, i.e., the amount of white noise at which the bound becomes trivial, of the analytical bound is rather small (similar to 0.06%), but the numerical method takes us into an experimentally relevant regime (similar to 5%). We conclude by investigating the amount of randomness that can be certified using these Bell violations. Perhaps surprisingly, we find that the qualitative behavior resembles the behavior of rigid inequalities such as the Clauser-Horne-Shimony-Holt inequality. This shows that rigidity is not strictly necessary for device-independent applications.