# Self-testing of binary observables based on commutation

## TL;DR

This paper introduces commutation-based measures for certifying binary quantum observables through Bell inequality violations, enabling robust self-testing and complete characterization of measurements and states in quantum systems.

## Contribution

It develops a family of measures based on commutation relations that can be estimated from Bell violations, providing tight trade-offs and complete self-testing results for qubit observables and multipartite states.

## Key findings

- Measures can be estimated from Bell violations in various scenarios.
- Maximal violation leads to complete self-testing and rigidity.
- Any pair of qubit projective observables can be robustly certified.

## 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 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 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 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.