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Self-testing of binary observables based on commutation

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In this talk we consider the problem of certifying binary observables based on a Bell inequality violation alone, a task known as self-testing of measurements. We introduce a family of commutation- based measures, which encode all the distinct arrangements of two projective observables on a qubit. These quantities by construction take into account the usual limitations of self-testing and since they are `weighted’ by the (reduced) state, they automatically deal with rank-deficient reduced density matrices. We show that these measures can be estimated from the observed Bell violation in several scenarios. The trade-offs turn out to be tight and, in particular, they give non- trivial statements for arbitrarily small violations. On the other extreme, observing the maximal violation allows us to deduce precisely the form of the observables, which immediately leads to a complete rigidity statement. In particular, we show that the n-partite Mermin- Ardehali-Belinskii-Klyshko inequality self-tests the n-partite Greenberger-Horne-Zeilinger state and maximally incompatible qubit measurements on every site for all n. Our results imply that any pair of projective observables on a qubit can be certified in a robust manner. Finally, we show that commutation-based measures give a convenient way of expressing relations between more than two observables. This talk is based on https://arxiv.org/abs/1702.06845 .

This talk is part of the CQIF Seminar series.

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