Beyond Wigner: Non-Invertible Symmetries Preserve Probabilities
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866911452217999360 |
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| author | Bartsch, Thomas Gai, Yuhan Schafer-Nameki, Sakura |
| author_facet | Bartsch, Thomas Gai, Yuhan Schafer-Nameki, Sakura |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_07110 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Beyond Wigner: Non-Invertible Symmetries Preserve Probabilities Bartsch, Thomas Gai, Yuhan Schafer-Nameki, Sakura Quantum Physics Strongly Correlated Electrons High Energy Physics - Phenomenology High Energy Physics - Theory Quantum Algebra 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. |
| title | Beyond Wigner: Non-Invertible Symmetries Preserve Probabilities |
| topic | Quantum Physics Strongly Correlated Electrons High Energy Physics - Phenomenology High Energy Physics - Theory Quantum Algebra |
| url | https://arxiv.org/abs/2602.07110 |