Beyond Wigner: Non-Invertible Symmetries Preserve Probabilities

TL;DR

This paper demonstrates that non-invertible symmetries in quantum theories can preserve probabilities by acting as trace-preserving channels between different Hilbert spaces, resolving a conflict with Wigner's theorem.

Contribution

It introduces a framework where non-invertible symmetries act as isometries between Hilbert spaces, extending the understanding of symmetry actions beyond unitary operators.

Findings

Non-invertible symmetries act as trace-preserving channels.

Symmetry defects connect different twisted Hilbert spaces.

Examples include Tambara-Yamagami, Fibonacci, and Yang-Lee symmetries.

Abstract

In recent years, the traditional notion of symmetry in quantum theory was expanded to so-called generalised or categorical symmetries, which, unlike ordinary group symmetries, may be non-invertible. This appears to be at odds with Wigner's theorem, which requires quantum symmetries to be implemented by (anti)unitary -- and hence invertible -- operators in order to preserve probabilities. We resolve this puzzle for (higher) fusion category symmetries $\mathcal{C}$ by proposing that, instead of acting by unitary operators on a fixed Hilbert space, symmetry defects in $\mathcal{C}$ act as isometries between distinct Hilbert spaces constructed from twisted sectors. As a result, we find that non-invertible symmetries naturally act as trace-preserving quantum channels. Crucially, our construction relies on the symmetry category $\mathcal{C}$ being unitary. We illustrate our proposal through several examples that include Tambara-Yamagami, Fibonacci, and Yang-Lee as well as higher categorical symmetries.