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We consider the problem of certifying binary observables based on a Bellinequality violation alone, a task known as self-testing of measurements. Weintroduce a family of commutation-based measures, which encode all the distinctarrangements of two projective observables on a qubit. These quantities byconstruction take into account the usual limitations of self-testing and sincethey are "weighted" by the (reduced) state, they automatically deal withrank-deficient reduced density matrices. We show that these measures can beestimated from the observed Bell violation in several scenarios and the proofsrely only on standard linear algebra. The trade-offs turn out to be tight and,in particular, they give non-trivial statements for arbitrarily smallviolations. On the other extreme, observing the maximal violation allows us todeduce precisely the form of the observables, which immediately leads to acomplete rigidity statement. In particular, we show that for all $n \geq 3$ the$n$-partite Mermin-Ardehali-Belinskii-Klyshko inequality self-tests the$n$-partite Greenberger-Horne-Zeilinger state and maximally incompatible qubitmeasurements on every party. Our results imply that any pair of projectiveobservables on a qubit can be certified in a truly robust manner. Finally, weshow that commutation-based measures give a convenient way of expressingrelations among more than two observables.

Self-testing of binary observables based on commutation