Principal-plane comparison

Diagonalize

The principal target modes ${h^{\prime }}_r=(H{R}_L{)}_r$ and captured modes ${f^{\prime }}_r=(F_L{R}_L{)}_r$ obey

For nonzero ${\theta }_r$, the $g_r$ are orthonormal and orthogonal to the target plane. Zero-angle terms are omitted.

Let $v_L$ be the passive one-particle rotation sending ${h^{\prime }}_r$ to ${f^{\prime }}_r$ in each principal plane and acting identically on its orthogonal complement. Let ${\mathcal{I}}_H$ select the target pair and discard the other participating modes. Then the captured-pair selector is

with both selected pairs identified with the same two memory coordinates. The operational input is the normal joint state ${\rho }_{{\mathfrak{v}}_{L}SR}$, not merely its target marginal.