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.

The paper asks how channel error composition and unequal arm noise affect the advantage of GHZ-based conference key agreement over pairwise BB84. It compares asymptotic secret fractions in a star-network model, charging pairwise distribution $N-1$ network uses versus one for an $N$-party GHZ state. Exact density-matrix simulations cover $N=3,4,5$ with independent dephasing, amplitude damping, or depolarizing noise on every arm. After normalization to average single-qubit channel infidelity, amplitude-damping and depolarizing crossover thresholds agree within 1.2% across these sizes. At $N=3$, their thresholds are approximately 0.03974 and 0.03975, versus 0.09472 for dephasing—a factor of 2.38 higher. All thresholds decrease with $N$. Stacking amplitude damping and depolarization onto dephasing lowers the dephasing-strength crossover from 0.48759 to 0.03488 when all three channel strengths are equal. The analytical explanation uses independent Pauli noise: the global phase correlation is $\prod_j(1-2g_j)$, whereas each dealer–receiver bit correlation is $(1-2f_1)(1-2f_i)$. These expressions yield a closed-form rate. An arithmetic–geometric-mean argument supports distributing rather than concentrating a fixed noise budget; dealer reassignment to the cleanest arm gives an 18% rate improvement in one asymmetric amplitude-damping example. Pauli-twirling amplitude damping shifts the $N=3$ crossover by 3.2% relative to exact simulation. The model excludes transmission loss, finite-key corrections, source and detector imperfections, memories, and implementation-security effects. The resource advantage depends on bottleneck-network accounting, and direct simulations stop at five parties; larger systems rely on the analytical expressions.