= Solution
The <beta barium borate> crystal supplies <spontaneous parametric down-conversion>: a weak <ultraviolet> pump can produce a pair of lower-frequency <photons>, with their frequencies and <wavevectors> selected by <conservation of energy> and <phase matching>. A suitably aligned and compensated source makes the two polarization alternatives coherent, giving the stated <Bell state>. The crystal alone does not guarantee that every collected pair has that state.
The two <polarizing beam splitters> route <horizontal polarization> and <vertical polarization> to different ports. Take $H$ from arm $a$ to the trigger $T$ and $V$ to arm $c$. The ordinary <beam splitter> in arm $b$ is a lossless, polarization-independent 50:50 splitter, with the unused input in the <vacuum state>. It splits <probability amplitudes>, rather than measuring which output a photon takes. The <half-wave plate> changes the relative phase of its principal polarizations by $\pi$; orient it so that a $V$ photon in $c$ becomes $(H+V)/\sqrt2$ in $c'$, up to a common phase. Finally, <photodetectors> absorb arriving <photons> and supply electrical clicks. They need not resolve <photon number> for the ideal four-photon, fourfold-coincidence calculation: a click at every detector already requires one photon at each.
\Image[/past-exam-of-the-laser-science-and-quantum-information-processing-course-of-the-university-of-oxford/2010/8-ghz-mode-network.png]
{title=Original mode diagram for photonic GHZ preparation by fourfold coincidence}
{description=The paths and polarization conventions used in the calculation. PBS denotes a <polarizing beam splitter>; BS is a polarization-independent 50:50 <beam splitter>. A trigger click alone does not guarantee the required one-photon occupation of all three other outputs.}
Choose the relative <wave phases> so that the final <polarizing beam splitter> sends $H_{c'}$ to $H_2$, $V_{c'}$ to $V_1$, $H_d$ to $H_1$ and $V_d$ to $V_2$. Such phase conventions must be used consistently: an uncompensated relative phase would give a phase-shifted <GHZ state>.
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