= Solution
In <entanglement swapping>, start with independent <Bell pairs> $|\phi^+\rangle_{12}|\phi^+\rangle_{34}$. Send <qubits> 2 and 3 to a central <Bell state analyzer>, retaining 1 and 4 at separate endpoints. Reordering the <tensor product> factors to group 23 and 14 gives the identity
$$
|\phi^+\rangle_{12}|\phi^+\rangle_{34}
=\frac12\left(
|\phi^+\rangle_{23}|\phi^+\rangle_{14}
+|\phi^-\rangle_{23}|\phi^-\rangle_{14}
+|\psi^+\rangle_{23}|\psi^+\rangle_{14}
+|\psi^-\rangle_{23}|\psi^-\rangle_{14}\right).
$$
Consequently each ideal <Bell-basis measurement> outcome has <probability> $1/4$ and projects 14 onto the corresponding <Bell state>, even though those two <photons> have never interacted. Communicating the two-<bit> result permits an endpoint <Pauli gate> to convert every outcome into the same chosen <Bell pair>. Without that record, the endpoints have $\rho_{14}=I_4/4$, a <product state>, rather than a known <entangled state>.
The optical <Bell state analyzer> overlaps <photons> 2 and 3 at a <balanced beam splitter>, with matched spatial modes, spectra and arrival times, and resolves the output <photon polarizations> using <polarizing beam splitters> and <single-photon detectors>. <Two-photon interference> maps <Bell states> to different output patterns. The antisymmetric $|\psi^-\rangle$ gives one <photon> in each output with opposite <polarizations>. The symmetric $|\psi^+\rangle$ gives two opposite-polarization <photons> in the same output; the two polarization detectors there can resolve this pattern. The $|\phi^\pm\rangle$ states give same-polarization pairs and remain mutually ambiguous in this basic setup. The detected pattern therefore heralds a particular endpoint <Bell state>. With ideal detection this arrangement succeeds unambiguously on two of four equally likely inputs, with <probability> $1/2$; the successful events can be selected. A complete analyzer would accept all four outcomes. The endpoint <quantum entanglement> is conditional on the measurement record, so it enables no instantaneous signalling.<Experimental Entanglement Swapping: Entangling Photons That Never Interacted by Jian-Wei Pan and others>[1998, experimental Bell projection and conditional endpoint correlations]{ref}
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