DrChinese said:
"The wave function enforces the correlations"? That's an answer? That isn't even a summary of an answer.
I disagree that it's not even a summary of an answer. And I have posted more than just that once sentence about it. But at least now you're acknowledging it and we can move on from there.
DrChinese said:
I'd like someone to walk us through the splitting and evolution of the systems point by point where they can be discussed.
Sure, I said I would take a stab at it and I will.
Let's first describe the overall setup to be sure we have it right. We have four photons. In the initially prepared state, photons 1 and 2 are entangled, and photons 3 and 4 are entangled. Both entanglements are maximal so by monogamy of entanglement there can't be any other entanglements involved.
In the middle of the experiment, so to speak, photons 2 and 3 are brought together and an experimental choice is made of whether or not to induce an entanglement swap. If a swap is induced, then after the swap (where "after" does not refer to time ordering, since as already noted it is possible to run this experiment where, for example, the photon 1 and 4 measurements are in the past light cone of the photon 2 and 3 swap/no swap decision--"after" only refers to logical ordering in terms of the analysis we are doing), photons 2 and 3 are entangled, and photons 1 and 4 are entangled. Again, both entanglements are maximal.
At the end of the experiment (which might, as noted, be in the past light cone of the "middle" described above, but it is logically the end for purposes of analysis), photons 1 and 4 are measured. To keep it simple, we will assume they are both measured along the same polarization axis, so if they are entangled, the measurements will always agree. If they are not entangled, there is an equal chance for the measurements to agree or disagree.
Now we can talk about how the MWI describes what happens to the wave function in the above. The starting wave function is, schematically (and ignoring normalization, which I will do throughout):
$$
\ket{\Psi}_0 = \ket{\psi}_{12} \ket{\psi}_{34} \ket{\text{ready}}
$$
where the lower case ##\psi## kets on the RHS are subscripted with the photons that are entangled in them, and the "ready" ket describes the state of the swap/no swap decision apparatus.
When photons 2 and 3 come together, either a swap happens or it doesn't. So after that decision is made (and in the MWI, the dynamics of that decision would be encoded in the Hamiltonian and would affect the wave function), the wave function becomes
$$
\ket{\Psi}_1 = \ket{\psi}_{14} \ket{\psi}_{23} \ket{\text{swap}} + \ket{\psi}_{12} \ket{\psi}_{34} \ket{\text{no swap}}
$$
In MWI-speak, we have had a split into two worlds: in the first, the swap happened and the entanglements are changed; in the second, the swap didn't happen and the entanglements are not changed.
When the photon 1 and 4 measurements are done, we have the final state, which will be:
$$
\begin{matrix}
\ket{\Psi}_2 = \left( \ket{\text{1 and 4 up}} + \ket{\text{1 and 4 down}} \right) \ket{\psi}_{23} \ket{\text{swap}} \\
+ \left( \ket{\text{1 and 4 up}} + \ket{\text{1 up and 4 down}} + \ket{\text{1 down and 4 up}} + \ket{\text{1 and 4 down}} \right) \ket{\psi}_2 \ket{\psi}_3 \ket{\text{no swap}}
\end{matrix}
$$
Here we have two more splits of worlds, in MWI-speak: on the "swap" side, we have a split into two worlds, in one of which photons 1 and 4 are both up and in the other they are both down; on the "no swap" side, we have a split into four worlds, corresponding to the four possible combinations of photon 1 and 4 results (since if there is no swap they are uncorrelated). (Note that I have assumed that no measurements are made on photons 2 and 3, so they just stay however they were after the swap/no swap decision is made and executed.)
Of course none of this says that the MWI is "local". The wave function itself, I would say (and Zeh says in the paper you referenced from him), is a nonlocal object, because it includes degrees of freedom that are spatially separated. But it's a perfectly good explanation of the correlations: as I said, they are enforced by the wave function, and more specifically by the possibilities that appear in the wave function. "Perfect correlation" between photons 1 and 4 if there is a swap just means the only possibilities that appear in the wave function if there is a swap are ones in which the measurement results on photons 1 and 4 agree.
