Pauli Error Composition Determines the Multipartite Advantage in Conference Key Agreement

Abstract

GHZ-based conference key agreement distributes a shared key to $N$ parties in one network use, whereas pairwise BB84 needs $N-1$. The survival of that advantage has been studied against party number, loss, and topology, but almost always with the channel fixed at depolarizing. Simulating both protocols exactly for $N=3,4,5$ under dephasing, amplitude damping, and depolarizing noise, we find the crossover is set by the channel's Pauli error composition, not its identity: normalized to average infidelity, amplitude damping and depolarizing agree to within about 1\% at each group size, while dephasing can tolerate roughly $2.4\times$ more at $N=3$. We derive a closed-form rate for arbitrary per-party Pauli noise and use it to prove that concentrating a fixed noise budget on one arm is strictly worse than spreading it, and that giving the dealer role to the cleanest arm improves the rate by roughly 18\%. One structural fact drives every result: the phase-error term is a product over all $N$ arms, while the bit-error term stays local to a pair.

12 pages, 6 figures, 3 tables