Good old quantum mechanics. There's only so much information in a wave function. If you start with a single pi0 meson, for example, it only has one polarization state. That single state might not be an eigenstate of the system, however: the single state might be a 50% superposition of two eigenstates, for example, so if you go to measure it, it will collapse into one of those eigenstates. If nothing perturbs the system, it will not collapse into an eigenstate; it will be happy in its superposed state. If it decays into two photons, they will inherit that superposition, but there is still no more information than you started out with: the polarization states of the photons can't be independent of each other.
Now, you might say that, OK, the polarization state of the pi0 meson collapsed into an eigenstate upon its decay; the polarizations of the photons are correlated, of course, but they were decided when the decay occurred; I can do just as well by preparing two independent photons with the same polarization, and putting them into the mirror boxes instead. You could say that, but you'd be wrong. The photons you prepare that way will satisfy Bell's Inequality when you look at an ensemble of correlations, whereas an ensemble of photons from pi0 decay will violate it. The collapse of the polarization state of the long-defunct pi0 doesn't occur until the first of the boxes is opened (even though the order of the openings is ambiguous).