Information-theoretic principle of emergent 1-form symmetries

TL;DR

This paper introduces an information-theoretic framework for understanding the emergence and loss of 1-form symmetries in quantum systems, linking symmetry emergence to long-range entanglement and quantum error correction.

Contribution

It proposes a novel information-theoretic characterization of emergent 1-form symmetries and develops methods to detect and analyze their transitions in complex quantum systems.

Findings

Analytically identifies regimes of 1-form symmetry emergence in lattice systems.

Demonstrates efficient detection of 1-form symmetries using quantum error correction.

Links symmetry breaking to topological quantum phase transitions.

Abstract

Higher-form symmetries act on sub-dimensional spatial manifolds of a quantum system. They can emerge as an exact symmetry at low energies even when they are explicitly broken at the microscopic level, making them difficult to characterize. In this work, we propose that the emergence of 1-form symmetries is information-theoretic in nature, precisely characterized by the preservation of information about how quantum states transform under the symmetry. As a consequence, the loss of the emergent 1-form symmetry is an information-theoretic transition which we argue to be revealed from the long-range entanglement in the ensemble of post-measurement states. We analytically determine the regimes in which a 1-form symmetry emerges in product states on one- and two-dimensional lattices. In analytically intractable regimes, we demonstrate how to efficiently detect 1-form symmetries with a global quantum error correction (QEC) decoder and numerically examine the information-theoretic transition of the 1-form symmetry, including systems with $\mathbb{Z}_2$ topological order. As a practical application of our framework, we show that once the 1-form symmetry is detected to exist, a topological quantum phase transition characterized by the spontaneous breaking of the 1-form symmetry can be accurately determined by a disorder parameter. We further argue that our proposed theory for emergent 1-form symmetries offers new perspectives on particle condensation and suggests sharp information-theoretic phase boundaries between Higgs and confining regimes in the $\mathbb{Z}_2$ lattice gauge theory.