  Quantum networks will provide multi-node entanglement over long distances to
enable secure communication on a global scale. Traditional quantum
communication protocols consume pair-wise entanglement, which is sub-optimal
for distributed tasks involving more than two users. Here we demonstrate
quantum conference key agreement, a quantum communication protocol that
exploits multi-partite entanglement to efficiently create identical keys
between N users with up to N-1 rate advantage in constrained networks. We
distribute four-photon Greenberger-Horne-Zeilinger (GHZ) states generated by
high-brightness, telecom photon-pair sources across up to 50 km of fibre,
implementing multi-user error correction and privacy amplification on resulting
raw keys. Under finite-key analysis, we establish $1.15\times10^6$ bits of
secure key, which are used to encrypt and securely share an image between the
four users in a conference transmission. We have demonstrated a new protocol
tailored for multi-node networks leveraging low-noise, long-distance
transmission of GHZ states that will pave the way forward for future multiparty
quantum information processing applications.
