Does local realism imply entangled photons are equal (or opposite)?

Question

I'm watching a video about Bell's inequality and how there can be no local hidden variables. They explain it using photons and whether they pass through a polarizer or not when they're oriented at different angles. I know there have been multiple attempts to disproof Bell's theorem over many years so I'm sure I'm missing something obvious here.

I understand that the idea is that if there were hidden variables, the photon would have 3 bits of information representing whether the photon will pass through the polarizer in each of the 3 possible orientations. Then they list every possible bit value combination which are 000, 001, 010 and so on. Then they imagine that two distinct random bits are read from the photon and a value is recorded indicating whether the two obtained values were the same or not. Finally, they show that there is no situation in which the 3 possible combinations of distinct bit indexes (1,2), (1,3) and (2,3) will yield less than 1/3 equal values: since bits have 2 possible values but 3 indexes are considered, there will always be at least 1 pair of equal bits.

The only assumption they supposedly make to reach the 1/3 result is local realism, which I don't fully understand but as far as I know it doesn't include the assumption that both photons have identical bit values. And if we don't assume this, then there should actually be 6 bits instead of 3: (000,000), (000,001) (000,010) and so on, where the left bits are the left photon's and the right bits are the right photon's. In this case there will be 6 possible combinations of distinct bit indexes since (a,b) and (b,a) are not the same. And if so there are cases in which there will be no equal values such as (000,111), while other combinations have 2/6, 2/6 and 6/6 equal values. By producing the right proportions of bit combinations the photons could end up yielding 25% total equal values, which AFAIK is weird but doesn't violate local realism.

What am I missing here? Is the video I'm watching a "dumbed down" version of the proof that doesn't really prove Bell's theorem?

Answer 1

Bell's theorem is based on two assumptions:

1. locality (what happens in one place cannot instantly influence what happens in some other, distant place)

2. statistical independence (the properties of the particles emitted by the source and the settings of the detectors are free/independent variables).

The conclusion of Bell's theorem is that a theory can only reproduce the predictions of quantum mechanics if it is non-local (assumption 1 is denied) or superdeterminismic (assumption 2 is denied).

Bohmian mechanics is an example of a theory that denies assumption 1. 't Hooft's cellular automaton interpretation, or Stochastic electrodynamics are theories denying the second assumption.

"Realism" is not an assumption here. You cannot reproduce QM with a local and non-realistic theory. No such theory can exist.

I would advise you to read Bell's original paper:

On the Einstein Podolsky Rosen paradox

https://cds.cern.ch/record/111654/files/vol1p195-200_001.pdf

The abstract reads:

"THE paradox of Einstein, Podolsky and Rosen was advanced as an argument that quantum mechanics could not be a complete theory but should be supplemented by additional variables. These additional variables were to restore to the theory causality and locality. In this note that idea will be formulated mathematically and shown to be incompatible with the statistical predictions of quantum mechanics. It is the requirement of locality, or more precisely that the result of a measurement on one system be unaffected by operations on a distant system with which it has interacted in the past, that creates the essential difficulty."

You can see from here that the introduction of hidden variables (what is usually implied by realism) was required in order to save locality. This is what EPR proved. Bell examines this path (the only one preserving locality). But you should not interpret this as implying that, somehow, denying realism helps locality. It does not.

Answer 2

Here are the issues to keep in mind.

1. You've done a pretty good job of addressing the 33% (1/3) vs 25% (1/4) arithmetic, which is one of the hardest to understand. That being for the case that the photons' hidden variables are exactly the same (or opposite, but let's ignore that for simplicity). So good job.

2. You are absolutely correct that in a hidden variables theory, the left and right photons potentially are not identical. In your example, that coincides with your (000,000), (000,001) (000,010) cases (and variations). In that case, you might be able to wiggle some changes to bring the hidden variables minimum down from 33% to closer to the actual of 25%. But...

3. There is also the requirement of "perfect" correlations with this scenario. Measuring both photons at the same angle settings - any same settings - always produces a matching (100%) result! That of course negates point 2. In your example, such cases as (000,001) or (000,010) can't/don't physically occur. You only see (000,000), (010,010), etc.

Answering your question then: Yes, entangled photon pairs are always equal (or opposite).

And in fact, it was the perfect correlations that were discovered first - in 1935, by Einstein and 2 others in the paper known as EPR. Bell discovered point 1 around 1964. They both must be met. So local realism cannot model the actual results of experiments.

A few more point. Around 1988, an important new paper was written introducing the GHZ Theorem. The also refutes local realism, and one of its authors won a Nobel for this and other work on entanglement. In GHZ, statistical probabilities are not involved. In every single GHZ run, Quantum Mechanics predicts the exact opposite of the local realistic assumption. Needless to say, experiment supports QM.

If you find this useful for anything, I also have a web page that loosely mirrors the video example you provided. It is called Bell's Theorem with Easy Math.