Generalising the Horodecki criterion to nonprojective qubit observables
Abstract
The Horodecki criterion provides a necessary and sufficient condition for a twoqubit state to be able to manifest Bell nonlocality via violation of the ClauserHorneShimonyHolt (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 twovalued 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 maximallymixed 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.
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 January 2022
 DOI:
 10.1088/17518121/ac44ee
 arXiv:
 arXiv:2109.09890
 Bibcode:
 2022JPhA...55d5301H
 Keywords:

 Bell nonlocality;
 Horodecki criterion;
 CHSH inequality;
 POVMs;
 compatible observables;
 Quantum Physics
 EPrint:
 J. Phys. A 55, 045301 (2022)