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Suppose the EPR* concept were true as it would be applied to entangled photon polarization. (Please note: I am not saying it is.) They thought QM was incomplete, because there must exist "elements of reality" (hidden variables) that supplied the highly correlated results on entangled particle pairs (label them A and B). By inference, we deduce that those results must in fact be predetermined - even if they are random. Of course Bell showed us that such an approach would require an appeal to non-local action, but that is not the issue I am asking about here. My questions are trying to get an idea of how much "hidden" information (if it existed) would be needed to actually provide results consistent with EPR (perfect correlations) and, to some "fair" degree, with the QM statistical predictions (Bell correlations). For our purposes, we are only modeling polarization and are ignoring momentum and any other observables.

If there were only a single (1) hidden variable - as Bell considered in his paper (Section III - "a unit vector") - we wouldn't get close enough to the quantum expectation value at many Alice/Bob angle pairs (although a single hidden variable does give exactly the right stats for perfect correlations). And polarized unentangled photons would not follow Malus' Law, which would be a major problem. We need something approximating the cos^2(theta) rule too. So we need more than 1 hidden variable to support this hypothetical approach, so I am wondering... how many? I don't believe I have ever seen a calculation on this.

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When we produce entangled photon pairs, we know that they will be perfectly correlated (or anti-correlated depending on PDC Type) at every angle when that same angle is used for both photons of a pair. Suppose Alice and Bob both set their polarizing beam splitters (PBS) to 0 degrees. We expect them to see both H or both V when looking at A and B. We set the PBS=14 degrees, we expect them to see both H or both V. We set the PBS=37 degrees, we expect them to see both H or both V. Ditto for all angles. Presumably there is randomness added to the process, as we never know whether the outcomes would be both H or both V.

Concerning A:

Suppose we ask: what is the smallest angle difference that can be discriminated in a Bell test? When we set Alice to 0 degrees and shift Bob from a setting of 0 degrees to 1 degree: the predicted match percentage drops from 100.00% to 99.97% - not much difference. Moving Bob's setting to 6 degrees (holding Alice at 0 degrees) yields an expectation of 98.91%, a difference of about 1%. That might be enough to discriminate in a Bell test.

Of course, there are settings in which even a single degree difference yields a much higher difference. Alice=0 degrees, Bob=45 degrees, you get a match of 50.00%. Change Bob by a single degree, to 46 degrees, and the match rate drops to 48.26% - a difference of nearly 2 degrees. I would say that would easily be detectible in a Bell test.

So do we need a new hidden variable every 6 degrees? Or a new hidden variable every 1 degree? Or a new hidden variable every 1/2 degree?

Concerning B:

Obviously, there is a relationship between 2 very close angle settings. I don't know if that makes the amount of total information required less or not. But clearly there must be a point at which a completely independent bit of information must be encoded, so that the randomness is apparent.

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So I might start as follows: I guess you'd need at least A=30 hidden variables containing polarization outcomes (spaced 3 degrees apart) in order to make a hidden variable approach viable (close enough to the QM predictions). B=2^30 permutations is about 1 billion. That is a bit less than the same number of permutations in the order of a single suit in a deck of cards (13! = 6 billion).

Hopefully you follow my example, let me know if the idea is not clear to you. Is any of this a reasonable approach to getting the answers for A and B? If so, I might conclude: Any entangled pair sporting "hidden variables" would be about as likely to match another randomly selected entangled pair as to 1 in B, whatever your B value comes to.

Thanks in advance.-DrC

*EPR: https://journals.aps.org/pr/pdf/10.1103/PhysRev.47.777

**So the EPR idea is that A and B contain the same set of hidden variables, and they are to be measured by Alice and Bob respectively and independently.**If there were only a single (1) hidden variable - as Bell considered in his paper (Section III - "a unit vector") - we wouldn't get close enough to the quantum expectation value at many Alice/Bob angle pairs (although a single hidden variable does give exactly the right stats for perfect correlations). And polarized unentangled photons would not follow Malus' Law, which would be a major problem. We need something approximating the cos^2(theta) rule too. So we need more than 1 hidden variable to support this hypothetical approach, so I am wondering... how many? I don't believe I have ever seen a calculation on this.

**So I am looking for the "back of a napkin" answer for these 2 numbers:**

A.How many (hidden) binary bits would be required to model entangled photon polarization, given that a typical BBo crystal produces entangled photon pairs with perfect correlations at any angle setting. We also need to be able to come "close" to the QM expectation value at other angles, but obviously there are no local hidden variable combinations that can exactly reproduce those (as Bell proved). For simplicity, let's only consider angles from 0 to 90 degrees. Would we need 90 hidden variables, one for each degree? (Obviously, there is nothing special about a degree itself.) Do you need a hidden variable for every 2 degrees, requiring 45? Or?A.

**B.**How many permutations of those are possible, assuming A bits are random. I might guess 2^A, assuming binary bits.-------------------------

When we produce entangled photon pairs, we know that they will be perfectly correlated (or anti-correlated depending on PDC Type) at every angle when that same angle is used for both photons of a pair. Suppose Alice and Bob both set their polarizing beam splitters (PBS) to 0 degrees. We expect them to see both H or both V when looking at A and B. We set the PBS=14 degrees, we expect them to see both H or both V. We set the PBS=37 degrees, we expect them to see both H or both V. Ditto for all angles. Presumably there is randomness added to the process, as we never know whether the outcomes would be both H or both V.

Concerning A:

Suppose we ask: what is the smallest angle difference that can be discriminated in a Bell test? When we set Alice to 0 degrees and shift Bob from a setting of 0 degrees to 1 degree: the predicted match percentage drops from 100.00% to 99.97% - not much difference. Moving Bob's setting to 6 degrees (holding Alice at 0 degrees) yields an expectation of 98.91%, a difference of about 1%. That might be enough to discriminate in a Bell test.

Of course, there are settings in which even a single degree difference yields a much higher difference. Alice=0 degrees, Bob=45 degrees, you get a match of 50.00%. Change Bob by a single degree, to 46 degrees, and the match rate drops to 48.26% - a difference of nearly 2 degrees. I would say that would easily be detectible in a Bell test.

So do we need a new hidden variable every 6 degrees? Or a new hidden variable every 1 degree? Or a new hidden variable every 1/2 degree?

Concerning B:

Obviously, there is a relationship between 2 very close angle settings. I don't know if that makes the amount of total information required less or not. But clearly there must be a point at which a completely independent bit of information must be encoded, so that the randomness is apparent.

-------------------------

So I might start as follows: I guess you'd need at least A=30 hidden variables containing polarization outcomes (spaced 3 degrees apart) in order to make a hidden variable approach viable (close enough to the QM predictions). B=2^30 permutations is about 1 billion. That is a bit less than the same number of permutations in the order of a single suit in a deck of cards (13! = 6 billion).

**What are the A and B values on your napkin?**Hopefully you follow my example, let me know if the idea is not clear to you. Is any of this a reasonable approach to getting the answers for A and B? If so, I might conclude: Any entangled pair sporting "hidden variables" would be about as likely to match another randomly selected entangled pair as to 1 in B, whatever your B value comes to.

Thanks in advance.-DrC

*EPR: https://journals.aps.org/pr/pdf/10.1103/PhysRev.47.777

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