Trade-off relations between measurement dependence and hidden information for factorizable hidden variable models

TL;DR

This paper investigates the relationship between measurement dependence and hidden information in factorizable hidden variable models, deriving new Bell inequalities that clarify the trade-offs and conditions for their realization.

Contribution

It introduces a measure of hidden information and derives relaxed Bell inequalities for models lacking measurement independence, revealing fundamental trade-offs.

Findings

Derived relaxed Bell-CHSH inequalities for factorizable models

Established a necessary and sufficient condition for hidden information measures

Revealed trade-off relations between measurement dependence and hidden information

Abstract

The Bell theorem is explored in terms of a trade-off relation between underlying assumptions within the hidden variable model framework. In this paper, recognizing the incorporation of hidden variables as one of the fundamental assumptions, we propose a measure termed `hidden information' taking account of their distribution. This measure quantifies the number of hidden variables that essentially contribute to the empirical statistics. For factorizable models, hidden variable models that satisfy `locality' without adhering to the measurement independence criterion, we derive novel relaxed Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) inequalities. These inequalities elucidate trade-off relations between measurement dependence and hidden information in the CHSH scenario. It is also revealed that the relation gives a necessary and sufficient condition for the measures to be realized by a factorizable model.