Long-range nonstabilizerness and phases of matter

TL;DR

This paper investigates long-range nonstabilizerness in many-body quantum states, especially ground states of gapped local Hamiltonians, revealing conditions for its presence and implications for phases of matter and quantum error correction.

Contribution

It provides a rigorous criterion for long-range nonstabilizerness in translation-invariant MPSs based on local tensors and mutual information quantization.

Findings

Long-range nonstabilizerness is generic in many-body states.

A sufficient condition for long-range nonstabilizerness in MPSs is derived.

Mutual information between distant regions in stabilizer fixed points is quantized.

Abstract

Long-range nonstabilizerness can be defined as the amount of nonstabilizerness which cannot be removed by shallow local quantum circuits. In this work, 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 systems and present rigorous results in the context of translation-invariant matrix product states (MPSs). 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 this fact is not changed after applying shallow quantum circuits. We also discuss possible ramifications in the classification of phases of matter and quantum error correction.