So... what you have here is a deterministic non-local hidden variable model which violates Bell Inequalities. The reduced probabilities at Bob's end might look random to him, but fundamentally the measurement outcomes are determined by Alice and Bob's measurement choices. All good.
You also know that any deterministic local hidden variable model must obey Bell Inequalities.
What I'm saying is that any local hidden variable model must obey Bell Inequalities. You cannot increase the value of S by relaxing determinism.
So actually it's kind of a distraction to bring in determinism. Either you have local hidden variables - which obey Bell Inequalities - or you allow non-local hidden variables - in which case Bell Inequalities can be violated. Locality is the key assumption.
So... what you have here is a deterministic non-local hidden variable model which violates Bell Inequalities. The reduced probabilities at Bob's end might look random to him, but fundamentally the measurement outcomes are determined by Alice and Bob's measurement choices. All good.
You also know that any deterministic local hidden variable model must obey Bell Inequalities.
What I'm saying is that any local hidden variable model must obey Bell Inequalities. You cannot increase the value of S by relaxing determinism.
So actually it's kind of a distraction to bring in determinism. Either you have local hidden variables - which obey Bell Inequalities - or you allow non-local hidden variables - in which case Bell Inequalities can be violated. Locality is the key assumption.