Understanding the classical expectation side of EPR/Bell inequalities
I like building educational models[1], and I’m collecting ideas for a quantum physics based one.
I’m wanting to make sure I’ve got my head around the classical side of the Bell inequalities, specifically related using the sort of EPR type system.
I know the original system used spin, but polarisations are easier to measure in reality, and more ‘relatable’ to everyday folk, so I’m wanting to use H & V polarisation instead.
A sketch of the setup I have in mind is here: http://imgur.com/gallery/tM4be
There’s an entangled photon source which emits a pair of photons with polarisations differing by 90 degrees. On <PERSON>’s side is a vertical polarising filter, and on <PERSON>’s side is a rotatable polariser offset by an angle theta. Both sides have single photon detectors and which we’ll call events A and B respectively.
The key idea is to compare the expectations from a classical universe, and a quantum one. If I’m understanding correctly, here’s the two scenarios:
Quantum expectations:
If $A$ detects, then $$P(B) = \cos^2(\theta)$$ and $$P(\text{not}\,\,B)=1-\cos^2(\theta)$$
If not $A$ then $$P(\text{not}\,\,B) = \cos^2(\theta)$$ and $$P(B)=1-\cos^2(\theta)$$
Classical expectations: The source makes a matched pair of photons, which are 90 degrees offset from each other, but we don’t actually know their orientation. 474
Say the ‘true’ angle of <PERSON>’s photon is $\phi$, and <PERSON>’s is $90+\phi$. Then for our detectors:
$$P(A) = \cos^2(\phi)$$
$$P(B) = \cos^2(90 + \phi + \theta)$$
From both of these systems, we can collect the set of statistics on $P(A\, \text{&}\,\,\text{not}\, B)$, $P(\text{not}\, A\, \text{&}\, B)$, $P(A \,\text{&}\, B)$, $P(\text{not}\,A\,\, \text{&}\, \,\text{not}\,B)$, and see the differences between the two models. In particular we should be able to see that no matter what value phi is set to, the probabilities for the quantum system do not match the classical one. (Except for when theta is set to certain values, such as 0, 45 or 90 degrees, which would make the difference disappear)
So, is what I’ve written above correct? Am I doing justice to the classical side? Basically I’m wanting to make sure I don’t straw-man the non-quantum system, and make an unfair comparison.
Thanks all
[1]Such as https://tinkerings.org/2016/10/06/the-collatz-o-matic/ 680