Photonic GHZ preparation by fourfold coincidence

A postselection scheme in which two polarization-entangled photon pairs pass through polarizing beam splitters, a half-wave plate and a nonpolarizing beam splitter. One photon at each of a trigger and three output detectors selects a GHZ state in the three output polarizations, provided optical mode matching removes distinguishable pair labels. All four detections are needed for the selected event; the trigger alone does not guarantee the three other output occupations.

The 1999 experiment used an approximately $200\mathrm{fs}$, $394\mathrm{nm}$ pump pulse and a beta barium borate crystal. Two polarization-entangled photon pairs supplied four photons. A trigger and three other photodetectors selected a GHZ state through fourfold coincidence counting. Narrow interference filters lengthened the photon coherence time to approximately $500\mathrm{fs}$, exceeding the pump duration and suppressing timing information about which pair supplied which photon. The optical mode matching was essential to coherence, not merely to routing photons.^[ref]

The observed state was selected by all four detections, which also measured the photons. A trigger alone did not demonstrate a guaranteed surviving three-photon output. This distinction connects postselection to what an apparatus actually supplies, rather than identifying a fourfold coincidence with an unconditional source.^[ref]

For a source with exactly two photons in each input arm, let $q$ be the probability of one horizontal and one vertical photon in the trigger arm. This sector alone supplies exactly one trigger photon. Splitting the other arm across its two output ports has probability $1/2$; the final polarizing beam splitter selects equal polarizations with probability $1/2$. The fourfold success probability is therefore $q/4$. The two-pair component of polarization-entangled down-conversion has $q=1/3$, giving $1/12$; independent pair bookkeeping has $q=1/2$, giving $1/8$. Pair-label distinguishability can also destroy the conditional GHZ state coherence and must be checked separately.