Long-Range Nonstabilizerness and Phases of Matter

Long-range nonstabilizerness can be defined as the amount of nonstabilizerness that cannot be removed by shallow local quantum circuits. We study long-range nonstabilizerness in the context of many-body quantum physics, a task with possible implications for quantum-state preparation protocols and implementation of quantum-error correcting codes. After presenting a simple argument showing that long-range nonstabilizerness is a generic property of many-body states, we restrict to the class of ground states of gapped local Hamiltonians. We focus on one-dimensional (1D) systems and present rigorous results in the context of translation-invariant matrix product states (MPSs), with our analysis extending to all gapped 1D phases under widely accepted physical assumptions. By analyzing the fixed points of the MPS renormalization-group flow, we provide a sufficient condition for long-range nonstabilizerness, which depends entirely on the local MPS tensors. Physically, our condition captures the fact that the mutual information between distant regions of stabilizer fixed points is quantized and remains so after applying shallow quantum circuits.