I’m trying to better understand the assumptions behind Bell’s theorem, in particular the factorization (or “local causality”) condition.
As I understand it, Bell inequalities rely on the idea that joint probabilities can be written as:
P(A,B∣a,b,λ)=P(A∣a,λ)P(B∣b,λ),
where λ represents underlying variables.
This is usually interpreted as a statement about locality and hidden variables.
However, I’m wondering about a slightly different angle.
Suppose that what we call “observables” are not direct functions of an underlying state, but instead come from a many-to-one mapping (i.e. different underlying configurations correspond to the same observable outcome).
In other words, observable states correspond to equivalence classes of more detailed configurations.
My question is:
In such a situation, is it still expected that Bell-type factorization should hold at the observable level?
Or could the many-to-one nature of this mapping itself prevent a factorized description, even if the underlying dynamics are local?
I’m not trying to challenge Bell’s theorem itself, but to understand whether its assumptions implicitly rely on observables being “fine-grained enough” (i.e. effectively injective with respect to the underlying variables).
Are there known results or discussions about this kind of coarse-graining effect?