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

Abstract: Heralded multipartite entanglement distribution is a key requirement for device-independent conference key agreement (DI-CKA) over lossy quantum networks. Although locally equivalent in the absence of loss, different single-rail photon-number encodings of Greenberger-Horne-Zeilinger (GHZ) states can exhibit substantially different loss tolerance. We show that computational-basis GHZ states, comprising a coherent superposition of vacuum and an $n$-photon component, enable detection-loophole-free parity-CHSH violations at markedly lower detection efficiencies than previously considered fixed-photon-number GHZ states, and derive exact analytical conditions for the critical detection efficiencies of both state classes. Motivated by this advantage, we introduce a star-network protocol using heterogeneous sources to directly herald vacuum-$n$-photon GHZ states with long-distance scaling $O(η<em>{\mathrm{c}}<sup>{n/2})$, where $η</em>{\mathrm{c}}$ is the channel transmittance. For four users, we characterize the heralded state under photon loss and show that tunable source parameters preserve genuine multipartite entanglement at any finite channel distance. For both ideal Pauli measurements and experimentally accessible displacement-based measurements, our protocol enables DI-CKA at detection efficiencies achievable with current photodetectors, while retaining key rates and communication distances comparable to those of previous heralded schemes. We discuss physical implementations and analyze an SPDC-based realization, showing that source-induced asymmetry can make measurement-role assignment in the parity-CHSH test crucial. These results identify photon-number encoding, source architecture, and measurement-role assignment as design parameters for loss-tolerant multipartite quantum networks and enhanced DI-CKA performance.