Efficient Heralding of Loss-Tolerant Photonic GHZ States for Device-Independent Conference Key Agreement over Long Distances

Multipartite entanglement is essential for device-independent conference key agreement (DI-CKA), but its practical implementation is severely limited by optical loss in quantum networks. While different single-rail photon-number encodings of GHZ states are locally equivalent in the absence of loss, they respond differently to channel attenuation. This paper investigates the loss tolerance of two GHZ state families: the vacuum-n-photon state and the fixed-photon-number state. The authors derive exact analytical thresholds for detection efficiency in parity-CHSH tests and propose a star-network protocol that uses a heterogeneous source architecture to directly herald vacuum-n-photon GHZ states.

The study reveals that vacuum-n-photon GHZ states enable detection-loophole-free parity-CHSH violations at significantly lower detection efficiencies than fixed-photon-number states. This advantage is most pronounced in the few-user regime relevant to near-term quantum networks. The proposed star-network protocol achieves this by combining different classes of local sources, allowing for efficient heralding with a favorable long-distance scaling of O(η^n/2), where η is the channel transmittance. For four users, the authors show that tunable source parameters allow genuine multipartite entanglement to be certified at any finite channel distance, with the protocol's reach limited by detector dark counts rather than fidelity degradation.

This work identifies photon-number encoding, source architecture, and measurement-role assignment as critical design parameters for loss-tolerant quantum networks. By utilizing vacuum-n-photon states, the protocol enables DI-CKA at detection efficiencies achievable with current technology. Furthermore, the authors demonstrate that their approach supports security certification using experimentally accessible displacement-based measurements, providing a more transparent and practical path toward long-distance device-independent conference key agreement compared to previous methods.