Self-testing of an unbounded number of mutually commuting local observables

Based on the optimal quantum violation of suitable Bell's inequality, the device-independent self-testing of state and observables has been reported. It is well-studied that locally commuting or compatible observables cannot be used to reveal quantum nonlocality. Therefore, the self-testing of commuting local observables cannot be possible through the Bell test. In this work, we demonstrate the self-testing of a set of mutually commuting local observables. Such certification has not hitherto been reported. We show that the optimal quantum violations of suitably formulated bilocality and n-locality inequalities in networks uniquely fix the observables of one party to be mutually commuting. In particular, we first demonstrate that in a two-input-arbitrary-party star network, two commuting local observables can be self-tested. Further, by considering an arbitrary-input three-party bilocal network scenario, we demonstrate the self-testing of an unbounded number of mutually commuting local observables.