Lately I was studying the Bell and CHSH inequalities on Wikipedia (it has proven to be a good source to get an quick idea about everything). The articles are detailed and even provide the core of the proof in a mathematical derivation that is easy to understand. But it leaves me still with a few questions.
- How exactly does the given proof exclude every possible local realistic theory? For me it seems that it only excludes possibility of the correlation sum to exceed 2 if the physical theory only models the entangled particles and the detectors (all with hidden information). But this seems to neglect the influence of space/vacuum between the detectors which the particles travel through.
For example one could assume a theory where the vacuum is filled with various classical fields that are influenced by the detector setup. This influence may be local i.e. Information may travel with the speed of light (so just like for classical force fields). So the vacuum could polarize in some way according to the detector setup long before the experiment even begins (thus fill the space with the information about the measured axis).
Now I can assume the created entangled particle pair is a highly instable state that collapses immediately (picks a definite spin orientation) after the particles are apart in such a theory. Let’s say the collapse is highly sensitive to initial conditions such that the polarized vacuum randomly picks which spin orientation is chosen and let’s say this happens always in the direction of one of the detectors. This way it seems I can get a theory that violates Bell mathematically maximally to the value of 4.
However is it considered non local since the detectors effectively influence the particles seemingly non-locally via the vacuum? The definition of locality referenced by wiki does not look like any nontrivial vacuum was allowed.
To exclude such a construction I would assume the CHSH setup would require to set the orientation of the detectors after the particles are created and just before they arrive. Or was that done already but it’s just not mentioned in Wikipedia? I’ve read something about an inter-detector communication loophole but it does not sound as if it covers it – although the solutions to close this loophole look similar I do not see how it considers a communication (filling the space) before the experiment begins.
- The second question is about the QM calculation or better said the operator used for the spin measurement. The spin operators act on the wave function in every part of the space but how is that a realistic representation of the detector’s measurement operators? After all each detector can only detect the spin of particles that hit it – so it also is a location detector of some sort, too.
Thus I’d expect a more adequate representation of the detectors to be an operator which eigenstates are a combination of the possible eigenstates of both operators. Thus I thought restricting the spin operators to the space they are actually measured makes sense. While it does not make much difference for a single particle measurement it totally seems to breaks any kind of entanglement (mostly because the wave function only collapses locally so the part of the wave function in the other detector does not experience any change by measuring the spin of the first particle).
So it seems the non-local nature of the spin measurement seems to be a core aspect of breaking Bell’s inequality.
This said my general question is: given a blueprint of a detector, how do I derive the corresponding measurement operator correctly? I never had much to do with experimental physics so I never studied how this is done rigorously. Since I couldn’t find anything specific in wiki, I thought this might be a good place to ask.