Ok, so I have done some more reading and a lot more thinking about this, and I still can't understand how there can be any quantum teleportation if the entanglement of photons B & C occurs after A & D have been measured. This is *not* what was reported experimentally in this
http://128.84.158.114/abs/quant-ph/0201134" (PRL 88, [2002] art. 017903) that we were discussing earlier. In that case, photons 1 & 2 (equivalent to B & C in Dr. Chinese's example) enter the fiber beam splitter and are entangled before 0 & 3 are measured. The only thing that is delayed is the measurement of the Bell state that 1 & 2 have been projected into, which does not reflect on the *fact* of their entanglement, only the measurement of the state. Dr. Chinese has claimed that this does not matter, and that the same teleportation would be observed if A & D are measured before B & C are entangled. I cannot see how this can be correct, and I have worked out some of my arguments mathematically below. Please let me know where my mistake lies, if there is one.
Paraphrasing equations 2 and 3 from the paper cited above, the total wavefunction is initially composed of two independent states, and can be written as:
\left|\Psi_{tot}\right\rangle = \left|\Psi^{-}_{AB}\right\rangle \otimes \left|\Psi^{-}_{CD}\right\rangle, where \left|\Psi^{-}_{xy}\right\rangle refers to the Bell state,
\left|\Psi^{-}_{xy}\right\rangle = \frac{1}{\sqrt{2}}\left[\left|H\right\rangle_{x}\left|V\right\rangle_{y} - \left|V\right\rangle_{x}\left|H\right\rangle_{y}\right],
and H and V refer to orthogonal polarization states. \Psi_{tot} can then be re-expressed in the basis of Bell states of the A/D and B/C pairs:
\left|\Psi_{tot}\right\rangle = \frac{1}{2}\left[\left|\Psi^{+}_{AD}\right\rangle \otimes \left|\Psi^{+}_{BC}\right\rangle - \left|\Psi^{-}_{AD}\right\rangle \otimes \left|\Psi^{-}_{BC}\right\rangle - \left|\Phi^{+}_{AD}\right\rangle \otimes \left|\Phi^{+}_{BC}\right\rangle + \left|\Phi^{-}_{AD}\right\rangle \otimes \left|\Phi^{-}_{BC}\right\rangle\right]
Therefore, at the moment when Charlie entangles B & C in his fiber beam splitter, the system is cast into one of the four states above, and this means that A & D must also be entangled. At some later point, the particular Bell state of B & C is measured at the detectors, and at that moment, Charlie knows which Bell state A & D are in as well.
This is all fine, but it only works if the first equation I wrote above is valid
when B & C become entangled. This is not true if the measurements on A and D have already occurred. Immediately after those measurements have occurred, the total state of the system is known, that is, it has been resolved into some element of the set of
separable states:
\left\{\left[\left|H\right\rangle_{A}\otimes\left|V\right\rangle_{B}\otimes\left|H\right\rangle_{C}\otimes\left|V\right\rangle_{D}\right],\:\:<br />
\left[\left|V\right\rangle_{A}\otimes\left|H\right\rangle_{B}\otimes\left|H\right\rangle_{C}\otimes\left|V\right\rangle_{D}\right],\:\:<br />
\left[\left|H\right\rangle_{A}\otimes\left|V\right\rangle_{B}\otimes\left|V\right\rangle_{C}\otimes\left|H\right\rangle_{D}\right],\:\:<br />
\left[\left|V\right\rangle_{A}\otimes\left|H\right\rangle_{B}\otimes\left|V\right\rangle_{C}\otimes\left|H\right\rangle_{D}\right]\right\}
(Note: I used the tensor product notation above to emphasize the separability, but it is just the 4 combinations: HVHV, VHHV, HVVH, VHVH)
So there is now no way to get from just one of these states to the case where there is entanglement between A & D. Note that B & C aren't entangled in this case either ... (otherwise it would be possible to generate entangled pairs from linearly polarized photons simply using beamsplitters).
Anyway, hopefully this makes my analysis and arguments clear. Have I made a deduction or math error somewhere? Note that at no point do I involve order of detection in my analysis, I only refer to the state of the system when the B & C photons enter the beamsplitter.
Finally, it is worth noting that in the paper I cited, the authors did not make the same claim that Dr. Chinese made in his post here. They claim that the space-time separation of the
detection events doesn't matter, which is consistent with what they tested in their experiment.
). 
