Quantum Game with Photons

I'll make an attempt here, though. I think the effect on level 4 is totally classical so it shouldn't take too much background.

1. The solution to a problem involving light is called a wavefunction, which is just a name assigned to functions that map position and time onto what the wave is doing at that position and time. I.E. for a sound wave, W(x: position, t: time) => p: air pressure.

2. The meat of a "wave equation" is essentially also a function, but it's higher-order. It maps wavefunctions onto wavefunctions, I.E. consider high-order function L, such that L(W: wavefunction) => X: wavefunction. The name for this map is an "operator," by the way.

3. We can set up operators L(W) so that they map all "true and physical" wavefunctions on to the zero-function, f(x,t) = 0. This contrivance is the job of physicists to design; so for our discussion let's just take it that for every physically possible W(x,t), it is the case that L(W(x,t)) = 0. (And vice-versa, every solution to that equation is physically possible.) The equation L(W) = 0 is called the "wave equation," by the way.

3. It is a property of L that L(W + Q) = L(W) + L(Q) for wavefunctions W and Q. This implies that if L(W) = 0 and L(Q) = 0, then L(W + Q) = L(W) + L(Q) = 0+0 = 0. Therefore, since physical possibility L(W) = 0, then we may conclude that the sum of any two physically possible wavefunctions W and Q is another physically possible wavefunction, W+Q.

4. Ignore time and look at my nice graph[0]. This illustrates adding two functions (there A(x) and B(x) ) which also happen to be solutions to the wave equation. Play around with the parameter p, (whose purpose is to let you select A to be one of many horizontally offset versions of itself) and see if you can make A + B do anything noteworthy.

[0] https://www.desmos.com/calculator/iwa3auxvuz

5. Hopefully in looking at my graph, you have noticed that for some values of p, A+B became flat everywhere. Now, I can finally explain what's going on with level 4. When the beamsplitter produces two beams from one, the two functions it shoots out have two different values of p, the dynamics detailed in [1]. If two beams are incident on the splitter, four beams will shoot out, and since some of them are overlapping in space they will add and you'll see the superposition effects. See my drawing [2]. (By the way, the parameter p is called phase, and the verb for the thing the waves to under superposition is called interference.)

[1] https://en.wikipedia.org/wiki/Beam_splitter#Phase_shift

[2] I've drawn it here. Please don't over-interpret it, I just sketched it in paint without a lot of attention to detail. https://imgur.com/a/7YLMa

Hopefully this is helpful and sheds some light on the underlying physics!