Beyond Wigner: Non-Invertible Symmetries Preserve Probabilities

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.