Given a Bell inequality, if its maximal quantum violation can be achieved only by a single set of measurements for each party or a single quantum state, up to local unitaries, one refers to such a phenomenon as self-testing. For instance, the maximal quantum violation of the Clauser-Horne-Shimony-Holt inequality certifies that the underlying state contains the two-qubit maximally entangled state and the measurements of one party (say, Alice) contains a pair of anti-commuting qubit observables. As a consequence, the other party (say, Bob) automatically verifies his set of states remotely steered by Alice, namely the assemblage, is in the eigenstates of a pair of anti-commuting observables. It is natural to ask if the quantum violation of the Bell inequality is not maximally achieved, are we capable of estimating how close the underlying assemblage is to the reference one? In this work, we provide a systematic device-independent estimation by proposing a framework called robust self-testing of steerable quantum assemblages. In particular, we consider assemblages violating several paradigmatic Bell inequalities and obtain the robust self-testing statement for each scenario. Our result is device-independent (DI), i.e., no assumption is made on the shared state and the measurement devices involved. Our work thus not only paves a way for exploring the connection between the boundary of quantum set of correlations and steerable assemblages, but also provides a useful tool in the areas of DI quantum certification. As two explicit applications, we show 1) that it can be used for an alternative proof of the protocol of DI certification of all entangled states proposed by Bowles et al. [Phys. Rev. Lett. 121, 180503 (2018)], and 2) that it can be used to verify all non-entanglement-breaking channels with fewer assumptions compared with the work of Rosset et al. [Phys. Rev. X 8, 021033 (2018)].
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