This question is related to Physics SE Objective or Subjective, but is not the same. There was roughly a 10:1 imbalance of answers (weighted by reputation points) to the effect that entanglement is subjective - but this question comes at the issue from a different perspective.
If my understanding is correct, the following statements are true:
Entanglement between a pair of particles is a state variable of the two-particle system, more or less like spin, polarization, or energy.
Entanglement can be measured only by measuring many such pairs that are known to be produced in precisely the same way, and observing correlations in the measurement statistics.
This leads to the present question:
If absolutely nothing can be known a priori or post priori about the process by which photon pairs are produced, is it possible for photons in the pairs to be entangled?
I've tried to imagine an experiment that could test to see if entanglement is indeterminate -- sort of a Bell's test of entanglement per se. It seems that the experiment would require a way to switch between two different situations: one in which it's possible to know whether or not a given pair of photons is entangled, and one in which it's not possible to know. And, the results of the experiment need to be different in the two cases (by analogy with the double-slit experiment in which any possibility of knowing which slit a photon goes through makes the interference pattern disappear). However, I haven't come up with a suitable experiment yet.
If the answer to the question is “yes”, then it seems that entanglement can't can't really be an indeterminate state variable of the two-photon system – that entanglement would at best be a “hidden variable”.
If the answer is “no”, then it seems that entanglement can't be an intrinsic property of a photon pair: that somehow information about the photon pair production event needs to be available (though not necessarily accessed) along with the photon measurement process in order for there to be entanglement.