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

Zhi Li1,2, Dongjin Lee1,3, and Beni Yoshida1

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Phys. Rev. X 15, 021090 – Published 11 June, 2025

DOI: https://doi.org/10.1103/pw12-kdjx

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.

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Popular Summary

Quantum entanglement plays a fundamental role in the fascinating world of topological phases, where particles with unusual properties—such as anyons and emergent fermions—arise from the collective behavior of simple building blocks such as quantum bits. A fundamental question is just how much entanglement is needed to sustain such phenomena. In this study, we provide a quantitative answer using the geometric entanglement measure, which captures how different a quantum state is from one with only short-range entanglement. Our main finding is that topological systems must possess entanglement that scales extensively with the system size, meaning it grows at least linearly as the system grows.

To reach this conclusion, we analyze systems that support emergent particles and symmetries known as anomalous 1-form symmetries. These symmetries force the system to be long-range entangled; in other words, no short-range-entangled state can mimic their properties. We also examine 2D quantum error-correcting codes, particularly stabilizer codes, and show that the entanglement in these systems must scale at least quadratically with the code distance. This links the robustness of these codes to the amount of entanglement they contain, revealing a direct connection between entanglement and the ability to correct errors.

These results deepen our understanding of how entanglement supports exotic phases and quantum error correction. They also provide a framework for investigating how long-range entanglement underpins other quantum many-body systems.

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Phys. Rev. X 15, 021090– Published 11 June, 2025
  • Received 5 June 2024
  • Revised 28 November 2024
  • Accepted 15 May 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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