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arXiv · 2602.07110

Beyond Wigner: Non-Invertible Symmetries Preserve Probabilities

Thomas Bartsch·Yuhan Gai·Sakura Schafer-Nameki
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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.

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Thomas Bartsch, Yuhan Gai, Sakura Schafer-Nameki. 2026-02-06. Beyond Wigner: Non-Invertible Symmetries Preserve Probabilities. https://arxiv.org/abs/2602.07110

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