This paper investigates quantum contextuality, a central nonclassical aspect of quantum mechanics, by employing the algebraic and topological structures of modular tensor categories. The analysis establishes that braid group representations constructed from modular categories, including the SU(2)k and Fibonacci anyon models, inherently produce state-dependent contextuality, as revealed by measurable violations of noncontextuality inequalities. The explicit construction of unitary representations on fusion spaces allows this paper to identify a direct structural correspondence between braiding operations and logical contextuality frameworks. The results offer a comprehensive topological framework to classify and quantify contextuality in low-dimensional quantum systems, thereby elucidating its role as a resource in topological quantum computation and advancing the interface between quantum algebra, topology, and quantum foundations.
This paper investigates quantum contextuality, a central nonclassical aspect of quantum mechanics, by employing the algebraic and topological structures of modular tensor categories. The analysis establishes that braid group representations constructed from modular categories, including the SU(2)k and Fibonacci anyon models, inherently produce state-dependent contextuality, as revealed by measurable violations of noncontextuality inequalities. The explicit construction of unitary representations on fusion spaces allows this paper to identify a direct structural correspondence between braiding operations and logical contextuality frameworks. The results offer a comprehensive topological framework to classify and quantify contextuality in low-dimensional quantum systems, thereby elucidating its role as a resource in topological quantum computation and advancing the interface between quantum algebra, topology, and quantum foundations.
We describe the mathematical theory of topological quantum computing with symmetry defects in the language of fusion categories and unitary representations. Symmetry defects together with anyons are modeled by G-crossed braided extensions of unitary modular tensor categories. The algebraic data of these categories afford projective unitary representations of the braid group. Elements in the image of such representations correspond to quantum gates arising from exchanging anyons and symmetry defects in topological phases of matter with symmetry. We provide some small examples that highlight features of practical interest for quantum computing. In particular, symmetry defects show the potential to generate non-abelian statistics from abelian topological phases and to be used as as a tool to enlarge the set of quantum gates accessible to an anyonic device.
Abstract Topological quantum computation employs two-dimensional quasiparticles called anyons. The generally accepted mathematical basis for the theory of anyons is the framework of modular tensor categories. That framework involves a substantial amount of category theory and is, as a result, considered rather difficult to understand. Is the complexity of the present framework necessary? The computations of associativity and braiding matrices can be based on a much simpler framework, which looks less like category theory and more like familiar algebra. We introduce that framework here.
We propose a novel mathematical framework, rooted in fiber bundle theory, to quantify the phenomenon of quantum contextuality—a fundamental aspect of quantum entanglement. To demonstrate its utility, we first apply the formalism to quantum models in the foundations of quantum physics that both capture and differentiate varying strengths of contextuality. By revealing the intrinsic geometric structure of contextuality, the framework establishes a direct link to its computational foundations, facilitating the formulation of a topology-based model of computation that encapsulates the semantics inherent to contextuality. The framework delivers a rigorous and potentially more profound mathematical characterization of quantum contextuality, while offering new modeling methodologies for quantum cognition.
FINDING: Anyon braiding forms the computational substrate for topological quantum computation, with non-Abelian statistics encoded in braid group representations rather than local qubit states. | MATH: Braid group Bn generators σi satisfy σiσi+1σi=σi+1σiσi+1 (Yang–Baxter equation); non-Abelian anyons carry unitary representations U(σi) with U(σi)U(σj)=U(σj)U(σi) for ∣i−j∣≥2. Fibonacci anyons: fusion rule τ×τ=1+τ, Hilbert space dimension grows as Fibonacci sequence Fn, golden ratio ϕ=(1+5)/2≈1.618 appears as asymptotic dimension growth rate. Topological protection arises from degeneracy of ground states on genus-g surfaces: dimension =kg for level-k Chern–Simons theory. | CONNECTION: The Fibonacci anyon fusion space dimension ratio Fn+1/Fn→1.618 — direct geometric harmony link. Braid group relations ar Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
We investigate whether the braid group representation on N anyons uniquely determines the modular data (S and T matrices) of a non-Abelian topological order, and whether this framework classifies all fusion rules for Majorana zero modes (MZMs) in 2D topological superconductors. We show that the local N-anyon braid group representation constrains the modular data up to finite gauge ambiguity but does not uniquely fix the global S-matrix without specifying the underlying spin structure. For MZMs, the braid generators satisfy the Ising anyon R-matrix with the asymptotic Hecke relation σi2=1 (up to phase), which rigidly fixes the local F and R symbols of the Ising category. Consequently, the fusion rules are strictly of Ising type (σ×σ=1+ψ), precluding alternative non-Abelian fusion rules. The T-matrix is determined by topological spins, but the full S-matrix depends on the spin structure of the underlying manifold, reflecting the super-modular structure of the theory. ## 1. Introduction Topological quantum computation exploits the non-Abelian statistics of anyons—quasiparticle excitations in two-dimensional systems whose braiding operations are represented by non-commuting unitary matrices on a degenerate ground-state manifold [1]. A central question is whether the local braiding data of N anyons suffices to reconstruct the full modular data of the underlying topological order (TO), or whether additional global information is required. This question has direct physical consequences for 2D topological superconductors (TSCs) hosting Majorana zero modes (MZMs), where the anyonic excitations are vortices binding MZMs and the fermionic quasiparticle is a transparent (local) fermion [2,3]. The modular data of a TO consists of the S-matrix, which encodes the mutual braiding statistics of anyons, and the T-matrix, which encodes the topological spins. Together, these matrices form a projective representation of t
Majorana zero modes (MZMs) localized on vortices of a two-dimensional topological superconductor carry a projective representation of the braid group, and the associated anyon theory is encoded in a modular tensor category whose fusion rules and modular data (S and T matrices) determine the topological order. We address two questions: whether the braid group representation on N anyons uniquely determines the modular data of the underlying non-Abelian topological order, and whether a representation-theoretic framework can classify all possible fusion rules for MZMs in 2D topological superconductors. We develop a classification pipeline that (i) fixes the local MZM algebra as the Clifford algebra generated by 2n Majorana operators, (ii) imposes the pentagon and hexagon coherence equations on candidate braided fusion structures, and (iii) filters candidates by modular-data consistency conditions, including the Verlinde formula and Frobenius–Schur indicators. As a fully worked example we compute, with explicit arithmetic, the modular S matrix and total quantum dimension for the Ising-type theory relevant to chiral p-wave superconductors, obtaining D=2 and verifying unitarity of S to machine precision. We then analyze metaplectic categories SO(N)2, which generalize the Ising/MZM fusion pattern, and show how Property F (finiteness of braid images) constrains the classification. We argue that the framework classifies MZM fusion rules up to gauging and extension operations, but that non-semisimple and non-unitary candidates escape the semisimple machinery; we state falsification criteria and open problems. ## 1. Introduction A two-dimensional topological superconductor (2D TSC) supports vortices that bind Majorana zero modes (MZMs): self-adjoint fermionic operators γi=γi† obeying the Clifford algebra {γi,γj}=2δij. Exchanging vortices i and j acts on the ground-state manifold by the operato Full text and updates: papers.qnfo.org/papers/braid-group-representations-modular-data-and-the-classification-of-majo-5099abb8/