Self-Testing Positive Operator-Valued Measurements and Certifying Randomness

TL;DR

This paper demonstrates self-testing of extremal qubit POVMs in a device-independent setting, certifying more local and global randomness than projective measurements through novel Bell inequalities.

Contribution

It introduces new Bell expressions for self-testing extremal qubit POVMs and certifies higher amounts of randomness, advancing device-independent quantum randomness certification.

Findings

Approximately 1.58 bits of local randomness certified.

About 2.27 bits of global randomness certified.

C'3 inequality outperforms others in randomness certification.

Abstract

In the device-independent scenario, positive operator-valued measurements (POVMs) can certify more randomness than projective measurements. This paper self-tests a three-outcome extremal qubit POVM in the X-Z plane of the Bloch sphere by achieving the maximal quantum violation of a newly constructed Bell expression C'3, adapted from the chained inequality C3. Using this POVM, approximately 1.58 bits of local randomness can be certified, which is the maximum amount of local randomness achievable by an extremal qubit POVM in this plane. Further modifications of C'3 produce C''3, enabling the self-testing of another three-outcome extremal qubit POVM. Together, these POVMs certify about 2.27 bits of global randomness. Both local and global randomness surpass the limitations certified from projective measurements. Additionally, the Navascu\'es-Pironio-Ac\'in hierarchy is employed to compare the lower bounds on global randomness certified by C3 and several other inequalities. As the extent of violation increases, C3 demonstrates superior performance in randomness certification.