Non-invertible symmetries act locally by quantum operations

TL;DR

The paper addresses how non-invertible symmetries act on local operators in quantum field theories and many-body systems. It argues that these actions are naturally described by quantum operations, i.e. completely positive maps, via a Stinespring representation $\mathcal{X}(O)=V^\dagger \pi_{\mathsf{X}\overline{\mathsf{X}}}(O) V$. The authors develop a general argument and illustrate it with the Kramers-Wannier duality in the 1D Ising chain, where the duality wall implements a CPTP map with a Kraus form $\mathcal{D}(O)=\sum_t K_t O K_t^\dagger$. This work bridges non-invertible symmetry structures with quantum information language, suggesting new tools for analyzing recovery maps, spectral constraints, and higher-form generalizations of non-invertible symmetries.

Abstract

Non-invertible symmetries of quantum field theories and many-body systems generalize the concept of symmetries by allowing non-invertible operations in addition to more ordinary invertible ones described by groups. The aim of this paper is to point out that these non-invertible symmetries act on local operators by quantum operations, i.e. completely positive maps between density matrices, which form a natural class of operations containing both unitary evolutions and measurements and play an important role in quantum information theory. This observation will be illustrated by the Kramers--Wannier duality of the one-dimensional quantum Ising chain, which is a prototypical example of non-invertible symmetry operations.

Figures (4)

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• Figure 4: Action of a non-invertible symmetry $\mathsf{X}$ on a local operator $O$, making it manifest that it is a Stinespring representation of a completely positive map.
• Figure 5: Action of a non-invertible symmetry $\mathsf{D}$ on a local operator $O$ implemented by the walls in the model of a one-dimensional spin chain.