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Fusion and Structure of Hilbert Space

Steven H. Simon

Published Sep 29, 2023• 0 citations
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Abstract

Abstract In this chapter we begin the detailed structure of anyon theories, or TQFTs. We introduce the concept of fusion and explain how it is related to the Hilbert space of the anyon systems. We emphasize the importance of locality as a principle for understanding anyons. Associativity is a strong constraint on possible particle fusion rules. As examples, we introduce both Fibonacci anyons and Ising anyons. We explain how the fusion rules of an anyon theory can be used to calculate the degeneracy of the ground state on arbitrary 2-manifolds.

DOI
10.1093/oso/9780198886723.003.0008

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Abstract In this chapter we begin the detailed structure of anyon theories, or TQFTs. We introduce the concept of fusion and explain how it is related to the Hilbert space of the anyon systems. We emphasize the importance of locality as a principle for understanding anyons. Associativity is a strong constraint on possible particle fusion rules. As examples, we introduce both Fibonacci anyons and Ising anyons. We explain how the fusion rules of an anyon theory can be used to calculate the degeneracy of the ground state on arbitrary 2-manifolds.
Abstract We explicitly work out the details of some simple examples anyon theories. We start with a fusion ring and using consistency conditions we build F-matrices, then the possible braidings (R-matrices). We consider Z2 fusion rules, Fibonacci fusion rules and Z3 fusion rules as simple cases. We consider Ising fusion rules, obtaining eight possible (modular) unitary braided theories. We then describe several more abelian theories, including the toric code, and general ZN anyons. These can be extended to describe all possible abelian braided theories in terms of a limited number of so-called “prime” theories.
Abstract TQFTs, or anyon theories, have a large amount of addition mathematical structure which we explore in this chapter. We show that the quantum dimension which defines the weight of a loop in the diagram algebra is the same as the quantum dimension that describes the size of the many-anyon Hilbert space. We define the S-matrix in terms of diagrams, and the T-matrix in terms of twists. We show how the S- and T-matrices are related to the modular group of diffeomorphisms of the torus, and related further to the notion of central charge. We discuss how the many constraints on TQFTs enables one to develop a table of all possible sufficiently “small” TQFTs given certain very general conditions. We give tables for both modular and super-modular theories. We introduce the Kirby Color or Omega Strand as an important tool.
Topological quantum computation encodes and manipulates information by exclusively employing anyons. To study the computational power of anyons we plan to look into their fusion and braiding properties in a systematic way. This will allow us to identify a Hilbert space, where quantum information can be encoded fault-tolerantly. We also identify unitary evolutions that serve as logical gates. It is an amazing fact that fundamental properties, such as particle statistics, can be employed to perform quantum computation. As we shall see below, the resilience of these intrinsic particle properties against environmental perturbations is responsible for the fault-tolerance of topological quantum computation. Anyons are physically realised as quasiparticles in topological systems. Most of the quasiparticle details are not relevant for the description of anyons. This provides an additional resilience of topological quantum computation against errors in the control of the quasiparticles. In particular, the principles of topological quantum computation are independent of the underlying physical system. We therefore do not discuss its properties in this chapter. The abstraction might create a conceptual vacuum as many intrinsic properties of the system might appear to be absent. For example, we shall not be concerned with the trapping and transport of anyons or with geometrical characteristics of their evolutions. In this chapter we treat anyons as classical fundamental particles, with internal quantum degrees of freedom, much like the spin. Moreover, we assume that we have complete control over the topological system, in terms of initial-state preparation and final-state identification.
Until recently, a careful derivation of the fusion structure of anyons from some underlying physical principles has been lacking. In Shi et al. (2020), the authors achieved this goal by starting from a conjectured form of entanglement area law for 2D gapped systems. In this work, we instead start with the principle of exchange symmetry, and determine the minimal prescription of additional postulates needed to make contact with unitary ribbon fusion categories as the appropriate algebraic framework for modelling anyons. Assuming that 2D quasiparticles are spatially localised, we build a functor from the coloured braid groupoid to the category of finite-dimensional Hilbert spaces. Using this functor, we construct a precise notion of exchange symmetry, allowing us to recover the core fusion properties of anyons. In particular, given a system of n quasiparticles, we show that the action of a certain n-braid βn uniquely specifies its superselection sectors. We then provide an overview of the braiding and fusion structure of anyons in the usual setting of braided 6j fusion systems. By positing the duality axiom of Kitaev (2006) and assuming that there are finitely many distinct topological charges, we arrive at the framework of ribbon categories.
This chapter introduces the mathematical framework, basic rules, and some key results of quantum theory. After a succinct overview of linear algebra and an introduction to complex Hilbert space, it investigates the correspondence between subspaces of Hilbert space and propositions, their logical structure, and how the pertinent probabilities are calculated. It discusses the mathematical representation of states, observables, and transformations, as well as the rules for calculating expectation values and uncertainties, and for updating states after a measurement. Particular attention is paid to two-level systems, or ‘qubits’, and the connection is made with experimental evidence about binary measurements. The properties of composite systems are discussed in detail, notably the phenomenon of entanglement. The chapter concludes with an investigation of conceptual issues regarding realism, non-contextuality, and locality, as well as the classical limit.