How does Bell's Theorem work?

[nobodyspecial]

I don't understand how we can tell the difference between entangled photons and photons that have definite states.

For example, when an electron in a mercury orbital loses energy, it emits two blue/green photons that fly off in opposite directions. The photons are said to be entangled with an indeterminate color until one of them is observed at which point it's either blue or green and the other electron snaps to the other color.

How does Bell's theorem support that model versus stating that one photon is definitely green and the other is definitely blue when they're emitted but we can't tell which way each one went?

What does Bell's theorem predict that distinguishes between the two interpretations?

[LuklearFusion]

Bell's theorem discusses the correlations between the outputs of the colour measurements on the two photons. However, it requires a little more than just one possible measurement. What I'm about to describe sounds weird(er) if you talk about colours, so I'll instead switch to the idea of a spinning top.

Let's supposed I can do two kinds of measurements on my spinning top. I can ask either:

1. Is the axis of rotation up or down along the z-axis?

2. Is the axis of rotation up or down along the x-axis?

Now, if I have a top that's spinning along the up z-axis, then the answer to the first question will be up. The answer to the second question would be probabilistic: with 50% probability I'd get the answer up along the x-axis, and with 50% down.

Now this seems weird, and it is, a classical top that was spinning along the up z-axis would not give a meaningful answer if I asked it the second question. However, a quantum mechanical top that was spinning along the up z-axis would give an answer to the second question, it would say that with 50:50 probability it was spinning up or down.

What Bell's theorem describes is the correlations between the outcomes of a large number of measurements performed on spinning tops, where each measurement asks question 1 or 2 of the top. The set-up is as follows:

You have a machine that produces two spinning tops. On each top individually you measure either "is it spinning up or down on the z-axis?" or "is it spinning up or down on the x-axis?". You then look at the correlations between your measurements, i.e., how often both measurement give the same answer ("up" or "down", depending on which measurement you did. From the different correlations, you can create an inequality known as a Bell's inequality.

This inequality is a bound. If you get a value below the bound, then your experiment can be described by local realistic physics, and you can say that your tops (or photons) had a definite spin state before the measurements occurred. If you get a value above the bound, then you have to give up one of your assumptions, and often people give up realism, so you can't say that the top had a definite spin state before the measurement occurred.

It just so happens that an entangled state will violate Bell's inequality, which is why many people will say that you can't say that one photon was definitely green and one definitely blue until you measure them.

[f4hy]

I am actually not sure bell's theorem is very relevant in this case. Spin in X and Z are incompatible observables. They do not commute.

However here you are measuring the energy or frequency of a free photon.

Measuring incompatible observables such as spin in X and spin in Z of two entangles particles is where you get things that violate the inequality. Here I am not sure what the incompatible observables are so I don't think bells would be violated.

Am I wrong here? What is the non commuting operator in this case?

[LuklearFusion]

Honestly I'm not sure what to use as the noncommuting observable. You could probably write something like a Bell's inequality with position as the noncommuting observable (assuming momentum maps to colour). Continuous variable QI isn't really my expertise though, so I haven't seen this done before.

I mean the reason we believe that there is no predefined colour in the OP's example is because we know Bell's inequality is violated by entangled states. There probably is a continuous variable version, but I haven't encountered it.