= Heralded Bell-pair projection with two polarization parity filters
{title2=$P_{0000}=1/64$}
Start from three normalized <Bell states> $\Phi^+_{12}\Phi^+_{34}\Phi^+_{56}$. A <polarizing beam splitter> parity filter $\Pi_Z^{25}$ followed by $X$-basis outcome $++$, and a second filter $\Pi_X^{46}$ followed by $Z$-basis outcome $00$, have effective bras $\langle\Phi^+|_{25}/\sqrt2$ and $\langle\Phi^+|_{46}/\sqrt2$. Their contraction leaves $\Phi^+_{13}/8$. The herald <probability> is therefore $1/64$ per ideal three-pair input, or $1/16$ conditional on both parity-routing filters having succeeded. This distinction prevents renormalizing away the optical <postselection> cost.
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