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

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