Entangling Atomic Quantum Memories Using High-Order Modulated Coherent States

High-order coherent-state modulation with reflective cavity-coupled atomic memories can distribute multiple heralded ebits per transmitted optical mode, but requires a collective midpoint POVM optimized for maximum distillable entanglement rather than minimum-error state discrimination. An SRM-inspired 16-QAM receiver reaches $1.75$ ebits per mode at $0.5$~dB end-to-end loss, $3.8$~dB below the half-link capacity benchmark. For 4-QAM, a variational POVM with more outcomes than constellation points improves the SRM rate by $4.0\mathrm{\%}$ yet remains below capacity, revealing measurement-design headroom. The advantage over single-qubit-per-mode midpoint protocols persists for per-interface loss $<0.2$~dB and calibrated phase error $<0.2\pi$.