  Quantum conference key agreement (QCKA) allows multiple users to establish a
secure key from a shared multi-partite entangled state. In a quantum network,
this protocol can be efficiently implemented using a single copy of a N-qubit
Greenberger-Horne-Zeilinger (GHZ) state to distil a secure N-user conference
key bit, whereas up to N-1 entanglement pairs are consumed in the traditional
pair-wise protocol. We demonstrate the advantage provided by GHZ states in a
testbed consisting of a photonic six-user quantum network, where four users can
distil either a GHZ state or the required number of Bell pairs for QCKA using
network routing techniques. In the asymptotic limit, we report a more than
two-fold enhancement of the conference key rate when comparing the two
protocols. We extrapolate our data set to show that the resource advantage for
the GHZ protocol persists when taking into account finite-key effects.
