= Solution
The <optical success probability of a fourfold GHZ source> depends on the photon-number statistics of the assumed two-pair input. Let $q$ be the probability that arm $a$ contains one $H$ and one $V$, given two pairs. The <beam splitter> then puts one of the two $b$ photons into each output with probability $1/2$, and the final polarization-parity selection succeeds with probability $1/2$. With ideal <photodetectors>,
$$
\boxed{P(\text{fourfold}\mid\text{two pairs})=\frac q4.}
$$
For the normalized, fully overlapping <two-pair component of polarization-entangled down-conversion> used above, $q=1/3$, so the squared norm of the accepted component gives \b[$P=1/12$]. Treating the two emissions instead as independently labelled <Bell pairs> gives the commonly used bookkeeping estimate $q=1/2$ and \b[$P=1/8$]. This is a different pair-statistics approximation; it must not be obtained by dropping labels from an input state without renormalizing it. Genuine <GHZ state> coherence additionally requires erasing information about the pair of origin.
Let $P_2$ be the two-pair probability per pulse and $f$ the pulse <repetition rate>. The coincidence rate is \b[$R=fP_2q/4$]. With the usual order-of-magnitude estimate $P_2\simeq p^2$, the supplied $f=10^6\,\mathrm{s^{-1}}$ and $p=10^{-3}$ give $fP_2\simeq1\,\mathrm{s^{-1}}$. The fully overlapping model gives \b[$R\simeq0.083\,\mathrm{s^{-1}}$], about one event per twelve seconds; independent-pair bookkeeping gives \b[$R\simeq0.125\,\mathrm{s^{-1}}$], about one per eight seconds. Thus the estimate is of order \b[$10^{-1}\,\mathrm{s^{-1}}$] under these idealizations.
An exact prefactor cannot follow from the single-pair probability alone. For example, the <pair-number statistics of polarization-entangled down-conversion> in a single collected mode pair give $P_2\simeq3p^2/4$ at weak gain, and hence \b[$R\simeq fp^2/16=0.0625\,\mathrm{s^{-1}}$] with $q=1/3$. Multiple temporal or spectral modes change those statistics. Real collection losses and detector efficiencies further reduce the rate; they are excluded by the ideal calculation.
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