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 \(75\%\) to \(79.62\%\) for \(d=4\), and from \(83.33\%\) to \(87.15\%\) for \(d=6\). With a representative finite detector saturation threshold, \(n_{\rm sat}=7\), the respective certified values remain \(78.84\%\) and \(86.71\%\). 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.