Generalising the Horodecki criterion to nonprojective qubit measurements

TL;DR

This paper extends the Horodecki criterion to include nonprojective qubit measurements, providing necessary and sufficient conditions for Bell nonlocality with noisy or weak measurements, and exploring measurement compatibility.

Contribution

It generalizes the Horodecki criterion to nonprojective measurements by characterizing two-valued qubit observables with fixed strengths and angles, and derives conditions for Bell violation.

Findings

Maximal CHSH values for unbiased measurements on arbitrary states

Conditions for CHSH violation with biased measurements

A simple necessary condition for qubit observable compatibility

Abstract

The Horodecki criterion provides a necessary and sufficient condition for a two-qubit state to be able to manifest Bell nonlocality via violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality. It requires, however, the assumption that suitable projective measurements can be made on each qubit, and is not sufficient for scenarios in which noisy or weak measurements are either desirable or unavoidable. By characterising two-valued qubit observables in terms of strength, bias, and directional parameters, we address such scenarios by providing necessary and sufficient conditions for arbitrary qubit measurements having fixed strengths and relative angles for each observer. In particular, we find the achievable maximal values of the CHSH parameter for unbiased measurements on arbitrary states, and, alternatively, for arbitrary measurements on states with maximally-mixed marginals, and determine the optimal angles in some cases. We also show that for certain ranges of measurement strengths it is only possible to violate the CHSH inequality via biased measurements. Finally, we use the CHSH inequality to obtain a simple necessary condition for the compatibility of two qubit observables.