Witnessing Short- and Long-Range Nonstabilizerness via the Information Lattice

We study nonstabilizerness on the information lattice, and demonstrate that noninteger local information directly indicates nonstabilizerness. For states with a clear separation of short- and large-scale information, noninteger total information at large scales $\Gamma$ serves as a witness of long-range nonstabilizerness. We propose a folding procedure to separate the global and edge-to-edge contributions to $\Gamma$. As an example we show that the ferromagnetic ground state of the spin-1/2 three-state Potts model has long-range nonstabilizerness originating from global correlations, while the paramagnetic ground state has at most short-range nonstabilizerness.