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Beyond Wigner: Non-Invertible Symmetries Preserve Probabilities

Thomas Bartsch, Yuhan Gai, Sakura Schafer-Nameki

TL;DR

The paper resolves the tension between Wigner’s probability-preserving view of quantum symmetries and non-invertible (generalised) symmetries by proving the Categorial Probability Preservation (CPP) theorem: in unitary fusion categories, a defect $A$ acts on a fixed twisted sector $X$ not as a unitary on a single Hilbert space but as a direct-sum isometry into an enlarged space consisting of all relevant twisted sectors. This is achieved by constructing bases of transition channels across simple outgoing sectors and showing that the action via these channels is trace-preserving when all sectors are included, effectively modelling the action as a quantum channel. The authors illustrate with concrete examples—Group, Tambara-Yamagami, Rep$(S_3)$, Fibonacci, and Yang-Lee—showing how unitary categories yield well-defined isometries, while a non-unitary case (Yang-Lee) violates CPP, underscoring the role of unitarity. Generalisations to higher dimensions and systems with boundaries are developed, including 2-group and 2-representation examples, demonstrating that the CPP framework extends to unitary fusion $(d-1)$-categories and yields trace-preserving quantum channels for the action of higher-codimension defects on twisted sectors.

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.

Beyond Wigner: Non-Invertible Symmetries Preserve Probabilities

TL;DR

The paper resolves the tension between Wigner’s probability-preserving view of quantum symmetries and non-invertible (generalised) symmetries by proving the Categorial Probability Preservation (CPP) theorem: in unitary fusion categories, a defect acts on a fixed twisted sector not as a unitary on a single Hilbert space but as a direct-sum isometry into an enlarged space consisting of all relevant twisted sectors. This is achieved by constructing bases of transition channels across simple outgoing sectors and showing that the action via these channels is trace-preserving when all sectors are included, effectively modelling the action as a quantum channel. The authors illustrate with concrete examples—Group, Tambara-Yamagami, Rep, Fibonacci, and Yang-Lee—showing how unitary categories yield well-defined isometries, while a non-unitary case (Yang-Lee) violates CPP, underscoring the role of unitarity. Generalisations to higher dimensions and systems with boundaries are developed, including 2-group and 2-representation examples, demonstrating that the CPP framework extends to unitary fusion -categories and yields trace-preserving quantum channels for the action of higher-codimension defects on twisted sectors.

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 by proposing that, instead of acting by unitary operators on a fixed Hilbert space, symmetry defects in 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 being unitary. We illustrate our proposal through several examples that include Tambara-Yamagami, Fibonacci, and Yang-Lee as well as higher categorical symmetries.
Paper Structure (15 sections, 5 theorems, 146 equations)

This paper contains 15 sections, 5 theorems, 146 equations.

Table of Contents

  1. Mathematical Background
  2. Fusion Categories
  3. Tube Categories
  4. Proof of Main Theorem
  5. Examples
  6. Group Symmetry
  7. Tambara-Yamagami Symmetry
  8. Rep(S3) Symmetry
  9. Fibonacci Symmetry
  10. Yang-Lee Symmetry
  11. Generalisations
  12. Systems with Boundary
  13. Higher Dimensions
  14. Example: 2-Group Symmetry
  15. Example: $2\mathsf{Rep}(({\mathbb{Z}}_2 \!\times\! {\mathbb{Z}}_2)[1]\rtimes {\mathbb{Z}}_2)$

Key Result

Theorem 1

Let $T: \mathcal{P}(\mathcal{H}) \to \mathcal{P}(\mathcal{H})$ be a transformation of the ray space The ray space of a Hilbert space $\mathcal{H}$ is defined to be the set $\mathcal{P}(\mathcal{H}) := (\mathcal{H}\backslash \lbrace 0 \rbrace)/\mathbb{C}^{\times}$ of equivalence classes of non-zero v for all $[\Phi],[\Psi] \in \mathcal{P}(\mathcal{H})$. Then, there is an (anti)unitary linear operat

Theorems & Definitions (5)

  • Theorem 1: Wigner, 1931
  • Theorem 2: CPP
  • Theorem 3: Higher CPP
  • Theorem 4
  • Theorem 5