How Much Entanglement Is Needed for Topological Codes and Mixed States with Anomalous Symmetry?

TL;DR

This paper quantitatively links the amount of entanglement, measured by geometric entanglement, to the emergence of anyons, fermions, and anomalous symmetries in topological and mixed quantum states, revealing fundamental entanglement thresholds.

Contribution

It introduces a rigorous framework connecting geometric entanglement with emergent anyons, fermions, and anomalous symmetries, including for mixed states, advancing understanding of topological quantum phases.

Findings

GEM scales linearly with system size for emergent anyons and fermions.

GEM of 2D stabilizer codes must be at least quadratic in code distance.

States with anomalous 1-form symmetries are necessarily long-range-entangled.

Abstract

It is known that particles with exotic properties can emerge in systems made of simple constituents such as qubits, due to long-range quantum entanglement. In this paper, we provide quantitative characterizations of entanglement necessary for emergent anyons and fermions by using the geometric entanglement measure (GEM), which quantifies the maximal overlap between a given state and any short-range-entangled states. For systems with emergent anyons, based on the braiding statistics, we show that the GEM scales linearly in the system size regardless of microscopic details. The phenomenon of emergent anyons can also be understood within the framework of quantum error correction (QEC). Specifically, we show that the GEM of any 2D stabilizer codes must be at least quadratic in the code distance. Our proof is based on a generic prescription for constructing string operators, establishing a rigorous and direct connection between emergent anyons and QEC. For systems with emergent fermions, despite that the ground state subspaces could be exponentially huge and their coding properties could be rather poor, we show that the GEM also scales linearly in the system size. Our analysis establishes an intriguing link between quantum anomaly and entanglement: A quantum state respecting anomalous 1-form symmetries, must be long-range-entangled and have large GEM. Our results also extend to mixed states, such as the $ZX$-dephased toric code, providing a provably nontrivial class of intrinsically mixed-state phases.