DrChinese said:
We agree on almost everything, including your "without thinking" part. You might consider re-thinking that, it's good to take the occasional opportunity to question your own (entrenched) position. Read on below...
Yes, it is true that photons [2,3] are observed to be very near (within a range of about 0.25 to 1.5 meters apart) to each other, so near that you cannot distinguish [2] from [3]. However: in half of the "chosen" cases, these photons are orthogonal and yet the (remote/distant/non-local) swap occurs.
The photons [2,3] interacted with the polarizing beam splitter and were then detected. I don't understand, what you mean by "orthogonal". Of course with some probability one of these photons turn out to be in orthogonal polarization states. What has this to do with their interactions with the beam splitter and the detectors? After all the state of these two photons simply is ##1/4 \hat{1} \otimes \hat{1}##. They are not entangled before the measurement. The purpose of this measurement is to select them in an entangled state. For each of the four possible entangled states then also the photon pair [1,4], which were never anywhere in causal/local contact, is in one of the four specific Bell states. This is however due to the entanglement of the pairs [1,2] and [3,4] and not due to any "spooky actions at a distance".
DrChinese said:
a) Can you explain how orthogonal photons manage to interact?
I don't understand this question. The photons interact with the beam splitter and the detectors, no matter in which polarization state they are found in this measurement. I don't know, what you mean by "interact" in connection with the orthonality of the measured polarization states. We don't discuss photon-photon scattering here. That's way too small to be detected in such an experiment. In fact it has only been detected very recently in ultra-relativistic ultra-peripheral Pb-Pb collisions at the LHC.
DrChinese said:
b) Can you explain how the outcomes of a measurement "here" (locally) cause the entanglement of distant photons?
As said above, that's due to the entanglement of the pairs [1:2] and [3:4] before the measurement. The two pairs among themselves are not entangled at all. In the initial state you have
$$\hat{\rho}=\hat{\rho}_{12} \otimes \hat{\rho}_{34}$$.
After making a Bell measurement on photons 2 and 3 and projecting the four-photon system in any state, where photons 2 and 3 were found in one of the four Bell states, the photons 1 and 4 are also projected to such a Bell state. There was no interaction at a distance. It was just selection of a sub-ensemble due to the local measurement on photons 2 and 3 and the original entanglement of photons 1 and 2 and photons 3 and 4. This correlation is already there with the original preparation. There's no need to envoke some "spooky actions at a distance", which contradict QED. QED itself is sufficient to explain the outcomes of this entanglement and thus the validity of the entanglement swapping protocol.
DrChinese said:
c) Can you explain how the entanglement of distant photons is "caused" long after those other photons are measured?
The same as answering b). It doesn't matter, when I select the sub-ensembles. The correlations needed to explain the entanglement of photons 1 and 4 after the local measurement on photons 2 and 3 were already there with the preparation of these four photons. The possibility of post-selection together with locality of QED demonstrates that there's no causal effect of the local measurement on photons 2 and 3 on the state of photons 1 and 4, because there's no causal connection between the latter photons with the measurement appartus used for the measurement on the former.
DrChinese said:
d) Can you explain at what time the [1] photon ceased being entangled with [2], and started being entangled with [4]? (Keeping in mind that the [4] photon need not even be in existence at any time you select.)
There's no temporal order in this. That's the whole point of this experiment! Your can only say: It's the split into subensembles, which come into being at the moment the pair [2,3] has been detecting as being in one of the four Bell states.
DrChinese said:
The thing I keep pointing out is the obvious contradiction between so-called local* interpretations** and the experimentally verified quantum rules, those being: entanglement of photons from independent sources, entanglement of photons outside a common light cone, entanglement via remote swapping, entanglement via delayed swapping, inability of orthogonal photons to interact, and of course Monogamy of Entanglement***. Perhaps the theoretical view espoused by
@vanhees71 needs refinement in light of overwhelming evidence to the contrary?
Locality is a property of theories, and in relativistic QFT it has a well-defined mathematical meaning, namely the microcausality constraint being fulfilled for all local observable operators. There's no interpretation involved here. I've no idea, what you mean by the "inability of orthogonal photons to interact". Of course, any photon can interact with the optical equipment with the corresponding probability. There's nothing the prevents such an interaction only because there's another photon being in an orthogonal polarization state.
DrChinese said:
*Despite the published conclusions of thousands of experimentalists demonstrating quantum non-locality, stating that their conclusions are orthodox QM.
The experimentalists demonstrate the validity of local relativistic QFTs again and again. It's the most successful theory ever.
Of course with QM, being a non-relativistic theory, there's no contradiction with actions at a distance. There's of course no microcausality constraint in non-relativistic Q(F)T. It's not even possible to formulate such a principle in non-relativistic physics.
DrChinese said:
** Or any viewpoint that claims the swap is merely correlation or coincidence, and does not represent a non-local change to the overall system under study.
I'm clearly in this camp, because that's what local relativistic QFT describes by construction.
DrChinese said:
***MoE: Which prevents photon [1] from being entangled with photons [2] and [4] simultaneously.
In none of the states (at the original preparation nor for the sub-ensembles after projection of photons [2,3] to one of the four Bell states) is photon 1 being entangled with photons 2 and 4 simultaneously. I never claimed such a thing.