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

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.