Beyond Wigner: Non-Invertible Symmetries Preserve Probabilities

• URL: http://arxiv.org/abs/2602.07110v1
• Date: Fri, 06 Feb 2026 19:00:00 GMT
• Title: Beyond Wigner: Non-Invertible Symmetries Preserve Probabilities
• Authors: Thomas Bartsch, Yuhan Gai, Sakura Schafer-Nameki,
• Abstract summary: Wigner's theorem requires quantum symmetries to be implemented by (anti)unitary -- and hence invertible -- operators in order to preserve probabilities.<n>We propose that instead of acting by unitary operators on a fixed Hilbert space, symmetry defects in $mathcalC$ act as isometries between distinct Hilbert spaces constructed from twisted sectors.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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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