Rényi Multi-entropy for Stabilizer States: A Finite Field Gauss Sum Problem

Abstract: Rényi multi-entropy probes multi-partite structure beyond bipartite spectra, but its behavior in general quantum states and the entanglement structure it captures remain poorly understood. We develop a finite field Gauss sum framework for calculating it in stabilizer states. Within this framework, the replica contraction reduces to a quadratic Gauss sum if and only if the state is locally unitarily equivalent to a stabilizer state. We obtain analytic expressions for the Rényi multi-entropy of all tree graph states and large families of loop graph states. In contrast to their flat bipartite Rényi spectra, stabilizer states generically have nonflat Rényi multi-entropy spectra.