An operator-algebraic formulation of self-testing

TL;DR

This paper introduces a new operator-algebraic definition of self-testing correlations, establishing equivalences with standard definitions under certain conditions, and extending applicability to commuting operator models.

Contribution

It provides a novel operator-algebraic framework for self-testing, showing equivalence with standard definitions for extremal correlations and extending to infinite-dimensional models.

Findings

New operator-algebraic definition of self-testing.

Equivalence with standard definitions for extremal correlations.

Extension of self-testing concepts to commuting operator models.

Abstract

We give a new definition of self-testing for correlations in terms of states on $C^*$-algebras. We show that this definition is equivalent to the standard definition for any class of finite-dimensional quantum models which is closed, provided that the correlation is extremal and has a full-rank model in the class. This last condition automatically holds for the class of POVM quantum models, but does not necessarily hold for the class of projective models by a result of Baptista, Chen, Kaniewski, Lolck, Man{\v{c}}inska, Gabelgaard Nielsen, and Schmidt. For extremal binary correlations and for extremal synchronous correlations, we show that any self-test for projective models is a self-test for POVM models. The question of whether there is a self-test for projective models which is not a self-test for POVM models remains open. An advantage of our new definition is that it extends naturally to commuting operator models. We show that an extremal correlation is a self-test for finite-dimensional quantum models if and only if it is a self-test for finite-dimensional commuting operator models, and also observe that many known finite-dimensional self-tests are in fact self-tests for infinite-dimensional commuting operator models.