Hidden Variables for Pauli Measurements

TL;DR

This paper explores hidden variable models for Pauli measurements in quantum information, revealing fundamental limitations related to contextuality and nonlocality using spectral graph theory.

Contribution

It introduces two classes of hidden variable assignments for Pauli measurements, analyzing their limitations and implications for quantum contextuality and nonlocality.

Findings

Partial hidden variable assignments are incomplete but consistent.

Contextual hidden variable assignments are complete but inconsistent.

Large amounts of contextuality and nonlocality can be achieved with Clifford gates and measurements.

Abstract

The Pauli measurements (the measurements that can be performed with Clifford operators followed by measurement in the computational basis) are a fundamental object in quantum information. It is well-known that there is no assignment of outcomes to all Pauli measurements that is both complete and consistent. We define two classes of hidden variable assignments based on relaxing either condition. Partial hidden variable assignments retain the consistency condition, but forfeit completeness. Contextual hidden variable assignments retain completeness but forfeit consistency. We use techniques from spectral graph theory to characterize the incompleteness and inconsistency of the respective hidden variable assignments. As an application, we interpret our incompleteness result as a statement of contextuality and our inconsistency result as a statement of nonlocality. Our results show that we can obtain large amounts of contextuality and nonlocality using Clifford gates and measurements.