Self-testing of binary observables based on commutation

Jędrzej Kaniewski
Phys. Rev. A 95, 062323 – Published 15 June 2017

Abstract

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 and the proofs rely only on standard linear algebra. The trade-offs turn out to be tight, and in particular, they give nontrivial 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 for all n3 the n-partite Mermin-Ardehali-Belinskii-Klyshko inequality self-tests the n-partite Greenberger-Horne-Zeilinger state and maximally incompatible qubit measurements on every party. Our results imply that any pair of projective observables on a qubit can be certified in a truly robust manner. Finally, we show that commutation-based measures give a convenient way of expressing relations among more than two observables.

  • Received 14 March 2017

DOI:https://doi.org/10.1103/PhysRevA.95.062323

©2017 American Physical Society

Physics Subject Headings (PhySH)

Quantum Information, Science & TechnologyGeneral Physics

Authors & Affiliations

Jędrzej Kaniewski*

  • QMATH, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark

  • *jkaniewski@math.ku.dk

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Issue

Vol. 95, Iss. 6 — June 2017

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