 Two-pair component of polarization-entangled down-conversion

For two collected spatial modes $a,b$, each with polarizations $H,V$, put $K^{†}=a_H^{†}{b}_V^{†}+a_V^{†}{b}_H^{†}$. The one-pair state is a Bell state. The normalized two-pair component isThis follows by expanding $(K^{†}{)}^2$ and using the bosonic creation operator identity $(a^{†}{)}^n|0⟩=\sqrt{n!}|n⟩$. The three orthogonal normalized Fock states have equal probability $1/3$. A tensor product of independently labelled pairs is a different multimode description; identifying its labels without renormalization changes its norm.