Abstract We continue the development of diagrammatics now considering diagrams living in 3 dimensions. We generalize the discussion of the R-matrix for braiding (non-identical) particles around each other, and discuss the Hexagon equations which provide a consistency condition for the R-matrices. As with the pentagon, Hexagon solutions are also rigid.
Abstract We continue the development of diagrammatics now considering diagrams living in 3 dimensions. We generalize the discussion of the R-matrix for braiding (non-identical) particles around each other, and discuss the Hexagon equations which provide a consistency condition for the R-matrices. As with the pentagon, Hexagon solutions are also rigid.
Investigating the transformation properties of the conformal blocks in the Z3 Potts model we derive some braid matrices. From the obtained braid matrices we explicitly show how the hexagon and pentagon identities are satisfied.
Braiding operators corresponding to the third Reidemeister move in the theory of knots and links are realized in terms of parametrized unitary matrices for all dimensions. Two distinct classes are considered. Their (non-local) unitary actions on separable pure product states of three identical subsystems (say, the spin projections of three particles) are explicitly evaluated for all dimensions. This, for our classes, is shown to generate entangled superposition of four terms in the base space. The 3-body and 2-body entanglements (in three 2-body subsystems), the 3-tangles and 2-tangles are explicitly evaluated for each class. For our matrices, these are parametrized. Varying parameters they can be made to sweep over the domain (0,1).Thus braiding operators corresponding to over- and under-crossings of three braids and, on closing ends, to topologically entangled Borromean rings are shown, in another context, to generate quantum entanglements. For higher dimensions, starting with different initial triplets one can entangle by turns, each state with all the rest. A specific coupling of three angular momenta is briefly discussed to throw more light on three body entanglements.
Exchanging particles on graphs, or more concretely on networks of quantum wires, has been proposed as a means to perform fault-tolerant quantum computation. This was inspired by braiding of anyons in planar systems. However, exchanges on a graph are not governed by the usual braid group but instead by a graph braid group. By imposing compatibility of graph braiding with fusion of topological charges, we obtain generalized hexagon equations. We find the usual planar anyon solutions but also more general braid actions. We illustrate this with Abelian, Fibonacci, and Ising fusion rules.
Braiding operators corresponding to the third Reidemeister move in the theory of knots and links are realized in terms of parametrized unitary matrices for all dimensions.Two distinct classes are considered.Their (nonlocal) unitary actions on separable pure product states of three identical subsystems (i.e., the spin projections of three particles) are explicitly evaluated for all dimensions.This, for our classes, is shown to generate entangled superposition of four terms in the base space.The 3-body and 2-body entanglements (in three 2-body subsystems), the 3 tangles, and 2 tangles are explicitly evaluated for each class.For our matrices, these are parametrized.Varying parameters they can be made to sweep over the domain (0, 1).Thus, braiding operators corresponding to over-and undercrossings of three braids and, on closing ends, to topologically entangled Borromean rings are shown, in another context, to generate quantum entanglements.For higher dimensions, starting with different initial triplets one can entangle by turns, each state with all the rest.A specific coupling of three angular momenta is briefly discussed to throw more light on three body entanglements.
Multiple equations in math, physics, quantum information, and elsewhere are referred to as "the" Yang-Baxter equation, in spite of being a broad family of equations. Most of the equations are nonlinear matrix equations, where the unknown variable is a matrix. This is the case for the so called braided, algebraic, and generalized forms of "the" equation, which are the primary focus of this dissertation. Finding solutions to the various forms of these equations has been the subject of much research. The equations in all their forms are largely considered intractable in high dimensions, and only in dimension 2 have the solutions been fully classified. We begin with an introduction to quantum computation, with a focus on the topological model and its connection to the braid group. Next, we introduce the braided, algebraic, and generalized forms of "the" Yang-Baxter equation. We provide a full classification of diagonal solutions to each form. In particular, we show that any diagonal matrix is a solution to the algebraic form in any dimension, and each instance of the braided and generalized forms only have diagonal solutions that are scalar multiples of the identity. We exploit the relationship between the algebraic and braided forms to construct a solution in any dimension that is applicable to topological quantum computation as a universal gate. The generalized form of the equation is parameterized by three natural numbers, (d,m,l), and we show that the only invertible solutions when l ≥ m are scalar multiples of the identity. We completely classify all solutions arising from an X-shaped ansatz for five different choices of (d,m,l), and provide a complete classification of X-shaped solutions to every odd dimensional braided equation, where there are no X-shaped solutions in any dimension. We fully classify permutation solutions to each instance of the braided and algebraic equations that can be written as a product of 3 or fewer transpositions. We show that the problem of classifying all invertible upper triangular solutions to the 4-dimensional algebraic Yang-Baxter equation can be split into 48 cases, and fully classify one of the cases.