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arXiv:2602.07110

Beyond Wigner: Non-Invertible Symmetries Preserve Probabilities

Thomas Bartsch, Yuhan Gai, Sakura Schafer-Nameki

quant-phcond-mat.str-elhep-phhep-thmath.QASubmitted Feb 6, 2026• Announced Feb 10, 2026• 0 citations• Revised Feb 17, 2026 (v2)
v1 · Feb 6, 2026v2 · Feb 17, 2026
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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.

Comments
4 pages + Supplementary Material, v2: references added

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Lattice non-invertible symmetries have rich fusion structures and play important roles in understanding various exotic topological phases. In this paper, we explore methods to generate new lattice non-invertible transformations/symmetries from a given non-invertible seed transformation/symmetry. The new lattice non-invertible symmetry is constructed by composing the seed transformations on different sites or sandwiching a unitary transformation between the transformations on the same sites. In addition to known non-invertible symmetries with fusion algebras of Tambara-Yamagami \mathbb Z_N×\mathbb Z_N ℤ N × ℤ N type, we obtain a new non-invertible symmetry in models with \mathbb Z_N ℤ N dipole symmetries. We name the latter the dipole Kramers-Wannier symmetry because it arises from gauging the dipole symmetry. We further study the dipole Kramers-Wannier symmetry in depth, including its topological defect, its anomaly and its associated generalized Kennedy-Tasaki transformation.

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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 C\mathcal{C}C by proposing that, instead of acting by unitary operators on a fixed Hilbert space, symmetry defects in C\mathcal{C}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 C\mathcal{C}C being unitary. We illustrate our proposal through several examples that include Tambara-Yamagami, Fibonacci, and Yang-Lee as well as higher categorical symmetries.
Non-invertible categorical symmetries have emerged as a powerful tool to uncover new beyond-Landau phases of matter, both gapped and gapless, along with second order phase transitions between them. The general theory of such phases in (1+1)d has been studied using the Symmetry Topological Field Theory (SymTFT), also known as topological holography. This has unearthed the infrared (IR) structure of these phases and transitions. In this paper, we describe how the SymTFT information can be converted into an ultraviolet (UV) anyonic chain lattice model realizing, in the IR limit, these phases and transitions. In many cases, the Hilbert space of the anyonic chain is tensor product decomposable and the model can be realized as a quantum spin-chain Hamiltonian. We also describe operators acting on the lattice models that are charged under non-invertible symmetries and act as order parameters for the phases and transitions. In order to fully describe the action of non-invertible symmetries, it is crucial to understand the symmetry twisted sectors of the lattice models, which we describe in detail. Throughout the paper, we illustrate the general concepts using the symmetry category \mathsf{Rep}(S_3) 𝖱 𝖾 𝗉 ( S 3 ) formed by representations of the permutation group S_3 S 3 , but our procedure can be applied to any fusion category symmetry.
The classification of phases using categorical symmetries has greatly expanded the landscape of gapped and gapless phases. So far, however, these developments have largely been restricted to phases with unitary (higher-)categorical symmetries over C\mathbb{C}C. In this work, we incorporate anti-unitary symmetries, such as time-reversal symmetry Z2T\mathbb{Z}_2^TZ2T​, and show that the relevant physical structures are naturally described by fusion categories over R\mathbb{R}R. A class of real fusion categories, which we call Galois-real fusion categories, provides the correct categorical model for anti-unitary symmetries. A simple example is the time-reversal symmetry Z2T\mathbb{Z}_2^TZ2T​ itself. We discuss the basic structures of real fusion categories and present a range of examples, including the group-theoretical categories (GT)ω(G^T)^ω(GT)ω and Rep(GT)\mathsf{Rep}(G^T)Rep(GT) associated to anti-linear groups GTG^TGT, as well as non-invertible time-reversal symmetries described by a real analogue of Tambara--Yamagami fusion categories. We then classify gapped phases enriched with anti-linear symmetries in terms of module categories over Galois-real fusion categories. We furthermore apply the categorical formulation to prove dualities (i.e. gauge or Morita equivalences) of anti-linear symmetries generated by gauging subgroups. Complementing this, we also develop a Symmetry Topological Field Theory (SymTFT) framework, in which Galois-real fusion categories arise as boundary conditions of a Z2T\mathbb{Z}_2^TZ2T​-enriched SymTFT. Morita equivalent anti-linear symmetries are shown to arise as different boundaries of the same Z2T\mathbb{Z}_2^TZ2T​-enriched SymTFT.
The Landau paradigm organizes phases of matter by the spontaneous breaking of invertible symmetries, with order parameters valued in representations of ordinary groups. A cluster of recent preprints in `cond-mat.str-el` demonstrates, through complementary microscopic and categorical analyses, that this framework is insufficient even in principle for a broad class of transitions: those governed by non-invertible symmetries, whose fusion rules do not close in a group but in a richer algebraic structure called a fusion category. This paper synthesizes six primary findings from that corpus into a candidate structural reading—explicitly a heuristic unification, not a derivation from a single formalism—of how categorical symmetry data organizes gapped phases, critical points, and order parameters beyond the Landau–Ginzburg–Wilson paradigm. The core thesis is as follows. (1) Gapped phases in 1+1d and 2+1d are classified not by which symmetry group is broken, but by which Lagrangian algebra in the Drinfel'd center of the symmetry category is preserved in the ground-state subspace. (2) Deconfined quantum critical points (DQCPs) arise generically when gauging an anomalous invertible symmetry produces a non-invertible dual, and the resulting transition is dual to spontaneous breaking of that non-invertible symmetry. (3) Twin phases—inequivalent gapped phases sharing identical anyon content and ground-state degeneracy—are the categorical signature of transitions that are intrinsically beyond Landau even after all possible gaugings. (4) These structures are not merely formal: they are realized in concrete lattice models, including Rydberg spin chains and spin-1/2 chains with D8D_8D8​ and H8H_8H8​ symmetry, where numerical DMRG and exact diagonalization confirm the predicted critical theories. The primary falsification path is numerical: a DMRG or quantum Monte Carlo study of the dimer-to-ferromagnet transition in the D8D_8D8​ spin chain should find central charge c≈1c \approx 1c≈1 and confirm that no local order parameter distinguishes the two sides after gauging—if either condition fails, the categorical classification is incomplete. Sources are drawn from `cond-mat.str-el` (six papers), with two `cond-mat.dis-nn` and two `cond-mat.mes-hall` papers noted but not incorporated into the main synthesis. --- Authorship: Saluca Agentic AI Research Team (Saluca LLC). AI-drafted from arXiv preprint corpus on the date in the filename. Cited arXiv preprints: 2605.27166, 2605.27193, 2605.27672, 2605.28688, 2605.28794, 2605.29745, 2605.29875, 2605.30421, 2605.31185, 2605.31540, 2605.31601, 2605.31602, 2605.31856, 2606.06343, 2606.06412, 2606.06466, 2606.06490
Non-invertible symmetries of a quantum field theory (QFT) are a natural generalization of unitary symmetries, but in which the product of operators does not satisfy a group multiplication law. We show that such symmetry operations on states define a collection of quantum gates for a parallel quantum computation scheme that includes post-selection / projection as a gate. Structures such as gate complexity and more geometric complexity measures generalize to this setting. We provide a class of distance / distinguishability measures that extend the standard notion of distance for Lie groups to both continuous and discrete non-invertible symmetries, as well as more general linear combinations of unitary quantum gates. We illustrate these considerations by computing the distance between non-invertible symmetries in some 4D and 2D QFTs. We find that the simple objects of a symmetry category can be highly complex computationally.
Lattice non-invertible symmetries have rich fusion structures and play important roles in understanding various exotic topological phases. In this paper, we explore methods to generate new lattice non-invertible transformations/symmetries from a given non-invertible seed transformation/symmetry. The new lattice non-invertible symmetry is constructed by composing the seed transformations on different sites or sandwiching a unitary transformation between the transformations on the same sites. In addition to known non-invertible symmetries with fusion algebras of Tambara-Yamagami \mathbb Z_N×\mathbb Z_N ℤ N × ℤ N type, we obtain a new non-invertible symmetry in models with \mathbb Z_N ℤ N dipole symmetries. We name the latter the dipole Kramers-Wannier symmetry because it arises from gauging the dipole symmetry. We further study the dipole Kramers-Wannier symmetry in depth, including its topological defect, its anomaly and its associated generalized Kennedy-Tasaki transformation.