Entanglement swapping: four clicks define the selected pair

Keep the four labels

Pan and colleagues’ 1998 experiment prepared two polarization-entangled photon pairs. One photon from each pair entered a joint analyzer. The remaining photons were tested through correlations conditioned on that analyzer’s coincidence. The verification record was therefore fourfold, not merely two clicks at the central splitter.

The pair-creation events used two passages of a pump pulse through the same crystal. Calling this two initially separate pairs does not imply two isolated lasers or independently operated source stations. The preparation geometry belongs in the historical claim, especially when comparing 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

For an original teaching calculation, label the input pairs one–two and three–four. Choose each as the equal positive superposition of both-horizontal and both-vertical polarization. Denote this normalized Bell state by the symbol below. A projection of the central photons onto the same Bell state leaves the outer pair in that state.

The right-hand amplitude is unnormalized. Its squared norm is one quarter, the ideal probability for this particular Bell outcome. Normalizing removes that factor from the selected state, but it cannot remove the probability from the acquisition ledger. Other Bell outcomes require their own labels or corresponding frame corrections.

{}_{23}\!\langle\Phi^{+}|\bigl(|\Phi^{+}\rangle_{12}\otimes|\Phi^{+}\rangle_{34}\bigr)=\tfrac12|\Phi^{+}\rangle_{14}

Read the evidence: J.-W. Pan, D. Bouwmeester, H. Weinfurter and A. Zeilinger (1998), Experimental Entanglement Swapping, PRL 80, 3891–3894 M. 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 distinguish the mixture

An equal incoherent mixture of both-horizontal and both-vertical pairs gives exactly the same equal-orientation counts as that positive Bell state in the horizontal–vertical basis. Thus an attractive correlation in that basis alone does not test the missing coherence.

Rotate both analyzers to the diagonal basis. The Bell state gives matching diagonal outcomes, while the incoherent mixture gives all four joint outcomes equally. This explicit counterexample explains why complementary-basis verification matters. A state reconstruction still assumes calibrated analyzers and a consistent selected ensemble.

Read the evidence: J.-W. Pan, D. Bouwmeester, H. Weinfurter and A. Zeilinger (1998), Experimental Entanglement Swapping, PRL 80, 3891–3894 D. 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

A successful pattern needs an input contract

The historical analyzer selected the antisymmetric Bell outcome using interference and a coincidence at different splitter outputs. Good spatial and temporal overlap, spectral filtering and suppressed extra-pair emission were part of making that pattern meaningful. A generic pair of clicks is not automatically a Bell projection.

In our ideal positive-pair calculation, suppose one thousand valid two-pair inputs reach an otherwise perfect analyzer accepting one Bell outcome. About 250 trials qualify before detector losses. If each of four required detectors independently succeeds with probability one half, the expected fourfold count drops to about 15.6. Conditional state quality and usable rate are separate quantities.

Read the evidence: J.-W. Pan, D. Bouwmeester, H. Weinfurter and A. Zeilinger (1998), Experimental Entanglement Swapping, PRL 80, 3891–3894

The ignored label restores the unsorted record

For the ideal two-pair input, averaging all four central Bell outcomes without corrections leaves the outer pair maximally mixed. Neither remote station can recognize the selected Bell ensemble from its own local clicks. The central outcome must be communicated and matched with the appropriate event identifiers.

This does not make arbitrary postselection a certification method. Keep the analyzer rule fixed, report discarded and missed events, and test additional bases. In particular, conditioning outer detections in a fourfold demonstration is not by itself an event-ready, loophole-free Bell trial. That stronger task has its own timing and loss requirements.

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, 4287–4290 J.-W. Pan, D. Bouwmeester, H. Weinfurter and A. Zeilinger (1998), Experimental Entanglement Swapping, PRL 80, 3891–3894

Separate the protocol from its relatives

Swapping transfers an entanglement relation using a joint measurement; teleportation transfers an unknown state using shared entanglement, a measurement and communicated correction information. Their algebra is closely related, but their declared inputs and verification questions differ.

The existing teleportation and single-qubit tomography Labs help inspect those ingredients. They do not reproduce the 1998 four-photon apparatus or its two-qubit verification. The next reading keeps the swapping inputs but moves the central choice later, asking which record changes and which merely receives a new conditional label.