Optimal dual-mode spin codes for photon loss

Photon loss is the dominant source of noise in bosonic systems and a fundamental limitation in optical and superconducting qubits. We identify numerically optimal codes in the fixed total-photon-number subspace of two oscillators that maximize entanglement fidelity under pure loss noise. Restricting to a fixed total photon number subspace enables experimentally feasible error detection through photon-number measurements. The resulting dual-mode codes generalize the dual-rail encoding to higher photon numbers and offer a compact setting for exploring number-conserving bosonic quantum error correction.