The check needs all four photons
Start with two entangled photon pairs and take one photon from each pair to a central analyzer. A suitable joint result lets the remaining two photons form an entangled pair, even though they never interacted directly. Pan and colleagues tested this entanglement-swapping idea in 1998. They checked the outer photons only when the central analyzer gave its selected result, so the demonstration used four detections together.
Both pairs came from one pump pulse passing twice through the same crystal. Calling them two initially separate pairs does not imply two isolated lasers or independently run sources. Any account of the historical result should state this preparation, especially when comparing it with later network demonstrations.
Read the evidence: J.-W. Pan, D. Bouwmeester, H. Weinfurter and A. Zeilinger (1998), Experimental Entanglement Swapping, PRL 80, 3891–3894
Project, then normalize
Label the two input pairs one–two and three–four. In our teaching calculation, each pair is an equal, positive superposition of both photons horizontal and both vertical. A superposition preserves the relative phase between those possibilities, so changing the analysis direction can make them interfere. The Phi-plus symbol in the expression denotes that normalized Bell state. Selecting the same Bell state for the central photons leaves the outer pair in it too.
The amplitude on the right is not normalized. Its squared norm is one quarter, the ideal probability of this particular Bell outcome. Normalizing removes that factor from the selected state, but the outcome still occurs in only one quarter of the trials. The other Bell outcomes need their own labels or matching frame corrections.
Read the evidence: J.-W. Pan, D. Bouwmeester, H. Weinfurter and A. Zeilinger (1998), Experimental Entanglement Swapping, PRL 80, 3891–3894M. Żukowski, A. Zeilinger, M. A. Horne and A. K. Ekert (1993), Event-Ready-Detectors Bell Experiment via Entanglement Swapping, PRL 71, 4287–4290
One basis cannot tell entanglement from a mixture
In the horizontal–vertical basis, an equal incoherent mixture of both-horizontal and both-vertical pairs gives exactly the same matching counts as the positive Bell state. A strong correlation in that basis alone therefore cannot show whether the coherence is there.
Rotate both analyzers to the diagonal basis. The Bell state gives matching diagonal outcomes, while the mixture gives all four joint outcomes equally often. This counterexample is why the check must use complementary bases. Even then, reconstructing the state assumes calibrated analyzers and a consistently selected set of events.
Read the evidence: J.-W. Pan, D. Bouwmeester, H. Weinfurter and A. Zeilinger (1998), Experimental Entanglement Swapping, PRL 80, 3891–3894D. F. V. James, P. G. Kwiat, W. J. Munro and A. G. White (2001), On the Measurement of Qubits — author manuscript, arXiv:quant-ph/0103121
What the Bell analyzer needs
The original analyzer picked out the antisymmetric Bell outcome by interference, using a coincidence between the two different outputs of a beam splitter. That signal meant something only with good overlap in space and time, spectral filtering and suppression of extra pairs. Two arbitrary clicks do not automatically make a Bell projection.
In our ideal calculation, suppose one thousand valid two-pair inputs reach an otherwise perfect analyzer that accepts one Bell outcome. About 250 trials qualify before detector losses. If each of the four detectors fires independently with probability one half, the expected fourfold count falls to about 15.6. How good the selected state is and how often you get it are separate questions.
Read the evidence: J.-W. Pan, D. Bouwmeester, H. Weinfurter and A. Zeilinger (1998), Experimental Entanglement Swapping, PRL 80, 3891–3894
Without the central result, the outer pair looks random
For the ideal two-pair input, averaging over all four central Bell outcomes without corrections leaves the outer pair maximally mixed. Neither remote station can pick out the selected Bell ensemble from its own clicks. The central result has to be sent to them and matched to the right events.
That does not make any postselection a valid way to certify entanglement. Keep the analyzer rule fixed, report discarded and missed events, and test more bases. A fourfold demonstration that conditions on the outer detections is not by itself an event-ready, loophole-free Bell test; that stronger task has its own timing and loss requirements.
Swapping transfers an entanglement relation to a new pair; teleportation transfers an unknown state using shared entanglement. Both depend on a joint measurement and its result sent by a classical channel, so the distant data can be interpreted together.
Read the evidence: M. Żukowski, A. Zeilinger, M. A. Horne and A. K. Ekert (1993), Event-Ready-Detectors Bell Experiment via Entanglement Swapping, PRL 71, 4287–4290J.-W. Pan, D. Bouwmeester, H. Weinfurter and A. Zeilinger (1998), Experimental Entanglement Swapping, PRL 80, 3891–3894