I Bell's Inequality => 4 entangled Photons impossible?

[ObjectivelyRational]

TL;DR Summary

Same reasoning which can be used to disprove hidden variables for 2 entangled photons implies that having 4 entangled photons is impossible.

Correlation between polarization measurements of entangled photons at angles less than 45 are greater than classically statistically possible. No set of hidden variables can be preordained to explain the 75% correlation of photon measurements at 30 degrees and complete anticorrelation of measurements at 90 degrees.

IF 4 Photons could be entangled (polarization state), one could set up 4 polarization detectors oriented at 0 (for reference), 90 degrees, and at two other angles between 0 and 90 (let's choose 30 and 60 for simplicity). The measurements would need to correlate 75% for the (0, 30) detectors, (30,60) detectors, and the (60,90) detectors, and the measurements of the (0, 90) detectors would have to completely anticorrelate.

In the same way no set of preordained "hidden" set of values could be generated to ensure the measurements worked out this way without the "action at a distance" for only 2 photons, no set of repeated measurements can be written down which satisfies these requirements for four entangled photons.

IS the above proof correct?

 

[DrChinese]

Summary:: Same reasoning which can be used to disprove hidden variables for 2 entangled photons implies that having 4 entangled photons is impossible.

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IF 4 Photons could be entangled (polarization state), one could set up 4 polarization detectors oriented at 0 (for reference), 90 degrees, and at two other angles between 0 and 90 (let's choose 30 and 60 for simplicity). The measurements would need to correlate 75% for the (0, 30) detectors, (30,60) detectors, and the (60,90) detectors, and the measurements of the (0, 90) detectors would have to completely anticorrelate.

In the same way no set of preordained "hidden" set of values could be generated to ensure the measurements worked out this way without the "action at a distance" for only 2 photons, no set of repeated measurements can be written down which satisfies these requirements for four entangled photons.

IS the above proof correct?

Well...

First, you are using the wrong formula for entangled particles N>2. When spin/polarization is involved, there is a constant total spin TS4 for the system of N particles. If you measure 1 of the N to be S1, then the other 3 are now entangled (as a system of 3) and their new total spin is TS3=(TS-S1). There is nothing limiting this result as you imagine - there are many quantum solutions. The same applies as we go from N=3 down to N=2 within that system of 4.

Second, producing N=4 photon entanglement has been experimentally accomplished (so much for your proof). And there is no specific limit on N. That includes experimental entanglement of systems of over a billion particles. To your specific point, here is entanglement of 4 photons.

https://arxiv.org/abs/1508.01480

Third, your argument is backwards regarding hidden variables. The point of the Bell argument is that there are NO classical/separable/hidden variable sets of outcomes that obey the quantum statistical predictions. That would still be true whether N=2 or N>2. The inequality would be more complex though. (The quantum predictions are fine though.)

And although it is specific to some quantum interpretations (which are best discussed in the PF subforum devoted to those): Action at a distance is not ruled out by quantum mechanics. The usual term to describe entanglement is "quantum nonlocality" which is not precisely the same thing as "action at a distance". A system of entangled particles is quantum nonlocal, but no one knows exactly how it works. Check out:

https://www.physicsforums.com/forums/quantum-interpretations-and-foundations.292/