Squeezing-enhanced Pairwise Fusion of Photonic Qudits

Pairwise fusion gates are linear-optical measurements that herald Bell projections onto two-rail subspaces of two $d$-rail single-photon qudits. Without ancillary input photons, these passive measurements succeed with probability $1-d^{-1}$, with all failures confined to the diagonal logical subspace. We show that identical single-mode squeezers applied to the $2d$ interferometer outputs before photon-number-resolving detection recover part of this structured failure sector. Photon-number parity preserves the successful off-diagonal fusion signatures, while selected all-even patterns yield POVM elements proportional to definite pairwise Bell projectors. We derive the exact logical-space POVM and prove that a diagonal pattern is accepted if and only if its photon-number-imbalance vector has exactly two nonzero components of equal magnitude. The resulting closed elliptic-integral expression increases the ideal success probability, for instance, from $75\mathrm{\%}$ to $79.62\mathrm{\%}$ for $d=4$, and from $83.33\mathrm{\%}$ to $87.15\mathrm{\%}$ for $d=6$. With a representative finite detector saturation threshold, $n_{\mathrm{s}\mathrm{a}\mathrm{t}}=7$, the respective certified values remain $78.84\mathrm{\%}$ and $86.71\mathrm{\%}$. These results establish active Gaussian processing as a method for recycling structured measurement failures without ancillary input photons, at the cost of $2d$ squeezing operations and a larger photon-number range at detection.