TY - JOUR

T1 - Self-testing of binary observables based on commutation

AU - Kaniewski, Jędrzej

PY - 2017/6/15

Y1 - 2017/6/15

N2 - We consider the problem of certifying binary observables based on a Bell inequality violation alone, a taskknown as self-testing of measurements. We introduce a family of commutation-based measures, which encodeall the distinct arrangements of two projective observables on a qubit. These quantities by construction takeinto account the usual limitations of self-testing and since they are “weighted” by the (reduced) state, theyautomatically deal with rank-deficient reduced density matrices. We show that these measures can be estimatedfrom the observed Bell violation in several scenarios and the proofs rely only on standard linear algebra. Thetrade-offs turn out to be tight, and in particular, they give nontrivial statements for arbitrarily small violations. Onthe 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 n 3 the n-partiteMermin-Ardehali-Belinskii-Klyshko inequality self-tests the n-partite Greenberger-Horne-Zeilinger state andmaximally incompatible qubit measurements on every party. Our results imply that any pair of projectiveobservables on a qubit can be certified in a truly robust manner. Finally, we show that commutation-basedmeasures give a convenient way of expressing relations among more than two observables.

AB - We consider the problem of certifying binary observables based on a Bell inequality violation alone, a taskknown as self-testing of measurements. We introduce a family of commutation-based measures, which encodeall the distinct arrangements of two projective observables on a qubit. These quantities by construction takeinto account the usual limitations of self-testing and since they are “weighted” by the (reduced) state, theyautomatically deal with rank-deficient reduced density matrices. We show that these measures can be estimatedfrom the observed Bell violation in several scenarios and the proofs rely only on standard linear algebra. Thetrade-offs turn out to be tight, and in particular, they give nontrivial statements for arbitrarily small violations. Onthe 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 n 3 the n-partiteMermin-Ardehali-Belinskii-Klyshko inequality self-tests the n-partite Greenberger-Horne-Zeilinger state andmaximally incompatible qubit measurements on every party. Our results imply that any pair of projectiveobservables on a qubit can be certified in a truly robust manner. Finally, we show that commutation-basedmeasures give a convenient way of expressing relations among more than two observables.

U2 - 10.1103/PhysRevA.95.062323

DO - 10.1103/PhysRevA.95.062323

M3 - Journal article

VL - 95

JO - Physical Review A - Atomic, Molecular, and Optical Physics

JF - Physical Review A - Atomic, Molecular, and Optical Physics

SN - 1050-2947

IS - 6

M1 - e1005751.

ER -