Non-invertible symmetries act locally by quantum operations

TL;DR

This paper reveals that non-invertible symmetries in quantum field theories and many-body systems act locally through quantum operations, connecting symmetry concepts with quantum information theory.

Contribution

It demonstrates that non-invertible symmetries operate via quantum operations, unifying symmetry actions with quantum information processes.

Findings

Non-invertible symmetries act as quantum operations on local operators.

The Kramers--Wannier duality exemplifies non-invertible symmetry as a quantum operation.

This perspective bridges symmetry actions with quantum information theory.

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.