Braid Matrices Research Articles

$$\ell _1$$-norm in three-qubit quantum entanglement constrained by Yang–Baxter equation

Usually the $\ell_2$-norm plays vital roles in quantum physics, acting as the probability of states. In this paper, we show the important roles of $\ell_1$-norm in Yang-Baxter quantum system, in connection with both the braid matrix and quantum entanglements. Concretely, we choose the 2-body and 3-body S-matrices, constrained by Yang-Baxter equation. It has been shown that for 2-body case, the extreme values of $\ell_1$-norm lead to two types of braid matrices and 2-qubit Bell states. Here we show that for the 3-body case, due to the constraint of YBE, the extreme values of $\ell_1$-norm lead to both 3-qubit $|GHZ\rangle$ (local maximum) and $|W\rangle$ (local minimum) states, which cover all 3-qubit genuine entanglements for pure states under SLOCC. This is a more convincing proof for the roles of $\ell_1$-norm in quantum mechanics.

Effects of braiding architectures on damage resistance and damage tolerance behaviors of 3D braided composites

The primary goal of this research is to present a systematic study on the damage resistance and damage tolerance behavior of three-dimensional (3D) braided composites. Therefore, the low velocity impact (LVI) and compression after impact (CAI) tests of 3D braided composites with eight different braiding architectures and a traditional two-dimensional (2D) laminated composite were carried out. The transient impact force response demonstrates that for 3D braided composites, fiber and matrix work as a whole to withstand impact loads and exhibit a progressive failure process, which is a very different phenomenon from the 2D laminates. The damage morphologies of 3D braided composites are obviously different, for 3D four-directional and 3D five-directional composites the damage patterns are in fusiform shape, and in the case of 3D six-directional and 3D seven-directional braided composites, damage patterns exhibit cross-shape. The energy absorption rates of 3D braided composites are 3.53–7.6% lower than 2D laminate, which means less impact energy has been dissipated by matrix and fiber damage. The comparison of LVI damage parameters and CAI shows that 3D5d-A has better damage resistance and damage tolerance.

Bi-directional regulatable mechanical properties of 3D braided polyetheretherketone (PEEK).

Fabricating adjustable mechanical properties of biomaterials at different level remains an important challenge in obtaining suitable bone substitutes with stiffness at micro scale and toughness at macro level. In this study, the changes in structure and properties of three dimension braided polyetheretherketone (3D-braided PEEK) after hot-pressed at 345 °C, 355 °C and 365 °C were investigated. The hyperfield 3D microscopy images and DSC revealed more compact macrostructure and lower crystallinity of 3D braided PEEK at higher hot-pressing temperature. The depth-sensing indentation data exhibited a reduction in elastic modulus of 3D braided PEEK with increasing the hot-press temperature, but its strength and toughness measured via universal material experiment machine increased. These results suggested that the micro-mechanics properties of braided PEEK matrix can be adjusted by altering its intrinsic crystal structure that governed by the hot-pressing temperature, whereas its macro-mechanics characteristics by changing its macrostructure that depended on textile structure. Present findings made a breakthrough in simultaneously regulating the micro- and macro-mechanics properties of PEEK in an opposite manner, which should have promising application in the development of customized orthopedic implants.

Meso-Level Finite Element Modeling Method for Mechanical Response of Braided Composite Tube with Gradient Structure in Axial Direction

In this paper, a novel numerical simulation framework using meso-scale finite element model is developed to predict the mechanical properties and damage mechanisms for one-layer biaxial braided composite tubes with gradient braided structures in axial direction, which involves braided reinforcements with an evolution of braiding angle. To bridge the relationships between the braiding procedure and geometric model, this paper develops an automatic algorithm that generates the geometric model with requiring braiding parameters as input parameters. The braided fabric model is generated by circularly arraying the yarn model, which is established by sweeping varying cross-sections along the centerline with the control of guidelines. The geometric model of matrix pockets is obtained by extracting the fabric model from the whole geometric model of composite tube. After that, a braiding yarn mesh in hexahedron format and co-node matrix mesh in tetrahedron format are generated from the assembled braided fabric and matrix pockets. The framework established in this paper is validated by comparison with the experimental results of the composite tubes with braiding angles from 36.6° to 49.6° subjected to quasi-static axial compression.

New type of solutions of Yang-Baxter equations, quantum entanglement and related physical models

Starting from the Kauffman-Lomonaco braiding matrix transforming the natural basis to Bell states, the spectral parameter describing the entanglement is introduced through Yang-Baxterization. It gives rise to a new type of solutions for Yang-Baxter equation, called the type-II that differs from the familiar solution called type-I of YBE associated with the usual chain models. The Majorana fermionic version of type-II yields the Kitaev Hamiltonian. The introduced ℓ1-norm leads to the maximum of the entanglement by taking the extreme value and shows that it is related to the Wigner’s D-function. Based on the Yang-Baxter equation the 3-body S-Matrix for type-II is explicitly given. Different from the type-I solution, the type-II solution of YBE should be considered in describing quantum information. The idea is further extended to ℤ3 parafermion model based on SU(3) principal representation. The type-II is in difference from the familiar type-I in many respects. For example, the quantities corresponding to velocity in the chain models obey the Lorentzian additivity rather than Galilean rule (u + v). Most possibly, for the type-II solutions of YBE there may not exist RTT relation. Further more, for ℤ3 parafermion model we only need the rational Yang-Baxterization, which seems like trigonometric. Similar discussions are also made in terms of generalized Yang-Baxter equation with three spin spaces .

Unitarity of the Modular Tensor Categories Associated to Unitary Vertex Operator Algebras, I

This is the first part in a two-part series of papers constructing a unitary structure for the modular tensor category (MTC) associated to a unitary rational vertex operator algebra (VOA). Given a rational VOA, we know that its MTC is constructed using the (finite dimensional) vector spaces of intertwining operators of this VOA. Moreover, the tensor-categorical structures can be described by the monodromy behaviors of the intertwining operators. Thus, constructing a unitary structure for the MTC of a unitary rational VOA amounts to defining an inner product on each (finite dimensional) vector space of intertwining operators, and showing that the monodromy matrices of the intertwining operators (e.g. braiding matrices, fusion matrices) are unitary under these inner products. In this paper, we develop necessary tools and techniques for constructing our unitary structures. This includes giving a systematic treatment of one of the most important functional analytic properties of the intertwining operators: the energy bounds condition. On the one side, we give some useful criteria for proving the energy bounds condition of intertwining operators. On the other side, we show that energy bounded intertwining operators can be smeared to give rise to (unbounded) closed operators. We prove that the (well-known) braid relations and adjoint relations for unsmeared intertwining operators have the corresponding smeared versions. We also give criteria on the strong commutativity between smeared intertwining operators and smeared vertex operators localized in disjoint open intervals of S1 (the strong intertwining property). Besides investigating the energy bounds condition, we also study certain genus 0 geometric properties of intertwining operators. Most importantly, we prove the convergence of certain mixed products-iterations of intertwining operators. Many useful braid and fusion relations will also be discussed.

Introduction to topological quantum computation with non-Abelian anyons

Topological quantum computers promise a fault tolerant means to perform quantum computation. Topological quantum computers use particles with exotic exchange statistics called non-Abelian anyons, and the simplest anyon model which allows for universal quantum computation by particle exchange or braiding alone is the Fibonacci anyon model. One classically hard problem that can be solved efficiently using quantum computation is finding the value of the Jones polynomial of knots at roots of unity. We aim to provide a pedagogical, self-contained, review of topological quantum computation with Fibonacci anyons, from the braiding statistics and matrices to the layout of such a computer and the compiling of braids to perform specific operations. Then we use a simulation of a topological quantum computer to explicitly demonstrate a quantum computation using Fibonacci anyons, evaluating the Jones polynomial of a selection of simple knots. In addition to simulating a modular circuit-style quantum algorithm, we also show how the magnitude of the Jones polynomial at specific points could be obtained exactly using Fibonacci or Ising anyons. Such an exact algorithm seems ideally suited for a proof of concept demonstration of a topological quantum computer.

Scalable measurements of tow architecture variability in braided ceramic composite tubes

AbstractThe performance of braided ceramic matrix composites has been shown to depend on the spatial arrangement of tows; therefore, a new class of tools is required to measure irregularities in the composite architecture for components with intricate geometries. We report a scalable and robust reconstruction technique built upon stereoscopic digital image correlation that is able to efficiently measure the position of tows in arbitrarily shaped composites. This method was applied to triaxially braided ceramic matrix composite tubes intended for use as nuclear fuel cladding, which revealed both long‐range “systematic” tow packing defects associated with the manufacturing process and short‐range “intrinsic” defects due to the braid architecture. These findings suggested that the character of tow spacing variation in braided composite tubes was substantially more complex than in planar woven composites. These measurements are expected to lead to improved processing of braided composites and to facilitate the design of statistically representative virtual specimens for finite element modeling.

High-speed infrared thermal imaging during ballistic impact of triaxially braided composites

Ballistic impact experiments were performed on triaxially braided polymer matrix composites to study the heat generated in the material due to projectile velocity and penetration damage. Triaxially braided (0/+60/−60) composite panels were manufactured with T700S standard modulus carbon fiber and two epoxy resins. The PR520 (toughened) and 3502 (untoughened) resin systems were used to make different panels to study the effects of resin properties on temperature rise. The ballistic impact tests were conducted using a single stage gas gun, and different projectile velocities were applied to study the effect on the temperature results. Temperature contours were obtained from the back surface of the panel during the test through a high speed, infrared thermal imaging system. The contours show that high temperatures were locally generated and more pronounced along the axial tows for the T700S/PR520 composite panels; whereas, tests performed on T700S/3502 composite panels, using similar impact velocities, demonstrated a widespread area of lower temperature rises. Nondestructive, ultrasonic C-scan analyses were performed to observe the failure patterns in the impacted composite panels and correlate the C-scan results with the temperature contours. Overall, the impact experimentation showed temperatures exceeding 252℃ (485°F) in both composites which is well above the respective glass transition temperatures for the polymer constituents. This expresses the need for further high strain rate testing with measurement of the temperature and deformation fields in order to fully understand the complex behavior and failure of the material and to improve the confidence in designing aerospace components with these materials.

ℓ1-norm and entanglement in screening out braiding from Yang–Baxter equation associated with [formula omitted] parafermion

The relationships between quantum entangled states and braid matrices have been well studied in recent years. However, most of the results are based on qubits. In this paper, we investigate the applications of 2-qutrit entanglement in the braiding associated with Z3 parafermion. The 2-qutrit entangled state |Ψ(θ)〉, generated by the action of the localized unitary solution R˘(θ) of YBE on 2-qutrit natural basis, achieves its maximal ℓ1-norm and maximal von Neumann entropy simultaneously at θ=π/3. Meanwhile, at θ=π/3, the solutions of YBE reduces braid matrices, which implies the role of ℓ1-norm and entropy plays in determining real physical quantities. On the other hand, we give a new realization of 4-anyon topological basis by qutrit entangled states, then the 9×9 localized braid representation in 4-qutrit tensor product space (C3)⊗4 is reduced to Jones representation of braiding in the 4-anyon topological basis. Hence, we conclude that the entangled states are powerful tools in analysing the characteristics of braiding and R˘-matrix.

Effect of temperature on the failure modes of a triaxially braided polymer matrix composite

The temperature effect on the tensile behavior of a triaxially braided composite was investigated in two different loading directions. The damage evolution was monitored and the results were explained with the help of experimental, analytical and numerical methods. The evolution in the constituent elastic properties was responsible for stress redistribution within the composite explaining the differences in crack initiation and Ultimate Tensile Strength for both directions and testing temperatures. The study also revealed the potential of quick and efficient analytical homogenization models to explain failure paths in textile composites.

Geometric defects in quantum Hall states

We describe a geometric (or gravitational) analogue of the Laughlin quasiholes in the fractional quantum Hall states. Analogously to the quasiholes these defects can be constructed by an insertion of an appropriate vertex operator into the conformal block representation of a trial wavefunction, however, unlike the quasiholes these defects are extrinsic and do not correspond to true excitations of the quantum fluid. We construct a wavefunction in the presence of such defects and explain how to assign an electric charge and a spin to each defect, and calculate the adiabatic, non-abelian statistics of the defects. The defects turn out to be equivalent to the genons in that their adiabatic exchange statistics can be described in terms of representations of the mapping class group of an appropriate higher genus Riemann surface. We present a general construction that, in principle, allows to calculate the statistics of $\mathbb Z_n$ genons for any "parent" topological phase. We illustrate the construction on the example of the Laughlin state and perform an explicit calculation of the braiding matrices. In addition to non-abelian statistics geometric defects possess a universal abelian overall phase, determined by the gravitational anomaly.

Matrix Algorithm for Braiding Simulation of Three-Dimensional Four-Step Braided Composites

This paper proposes a new kind of computational approach of three-dimensional (3D) braiding simulation based on matrix theory and symbol operation, which is suitable for batch integral calculation. The approach can be used to calculate braiding matrix at any step. Different braiding matrices fully reflect yarn carrier position changes. Using the braiding matrix calculation, the approach can provide data for storing yarn carrier paths, also termed braiding paths. The 3D braiding paths study was conducted to better visualization and thereby analyse the carrier’s movement characteristics and principles of operation. Finally, a program was designed to provide a reference for the mathematical model in further research.

On some 3-point functions in the W 4 CFT and related braiding matrix

We construct a class of 3-point constants in the $sl(4)$ Toda conformal theory $W_4$, extending the examples in Fateev and Litvinov. Their knowledge allows to determine the braiding/fusing matrix transforming 4-point conformal blocks of one fundamental, labelled by the 6-dimensional $sl(4)$ representation, and three partially degenerate vertex operators. It is a $3 \times 3$ submatrix of the generic $6 \times 6$ fusing matrix consistent with the fusion rules for the particular class of representations. We check a braiding relation which has wider applications to conformal models with $sl(4)$ symmetry. The 3-point constants in dual regions of central charge are compared in preparation for a BPS like relation in the $\hat{sl}(4)$ WZW model.

Conley index and tubular neighborhoods II

This paper is a sequel to our previous paper [11]. We prove existence of nested index filtrations for tubular singular semiflow perturbations. This generalizes a corresponding result from [7] and is applicable to cases in which the phase space of the perturbed semiflow is not necessarily homeomorphic to a product of metric spaces having as a factor the phase space of the limiting semiflow. This result implies tubular homology index braid and connection matrix continuation principles.

HOMFLY Polynomial from a Generalized Yang–Yang Function

Starting from the free field realization of Kac–Moody Lie algebra, we define a generalized Yang–Yang function. Then for the Lie algebra of type $$A_{n}$$ , we derive braiding and fusion matrix by braiding the thimble from the generalized Yang–Yang function. One can construct a knots invariant H(K) from the braiding and fusion matrix. It is an isotropy invariant and obeys a skein relation. From them, we show that the corresponding knots invariant is HOMFLY polynomial.

Kauffman Polynomial from a Generalized Yang–Yang Function

For the fundamental representations of the simple Lie algebras of type B n , C n and D n , we derive the braiding and fusion matrices from the generalized Yang–Yang function and prove that the corresponding knot invariants are Kauffman polynomial.

Colored HOMFLY polynomials of knots presented as double fat diagrams

Many knots and links in S^3 can be drawn as gluing of three manifolds with one or more four-punctured S^2 boundaries. We call these knot diagrams as double fat graphs whose invariants involve only the knowledge of the fusion and the braiding matrices of four-strand braids. Incorporating the properties of four-point conformal blocks in WZNW models, we conjecture colored HOMFLY polynomials for these double fat graphs where the color can be rectangular or non-rectangular representation. With the recent work of Gu-Jockers, the fusion matrices for the non-rectangular [21] representation, the first which involves multiplicity is known. We verify our conjecture by comparing with the [21] colored HOMFLY of many knots, obtained as closure of three braids. The conjectured form is computationally very effective leading to writing [21]-colored HOMFLY polynomials for many pretzel type knots and non-pretzel type knots. In particular, we find class of pretzel mutants which are distinguished and another class of mutants which cannot be distinguished by [21] representation. The difference between the [21]-colored HOMFLY of two mutants seems to have a general form, with A-dependence completely defined by the old conjecture due to Morton and Cromwell. In particular, we check it for an entire multi-parametric family of mutant knots evaluated using evolution method.

Quantum logic gates from Dirac quasiparticles

We show that one of the fundamental operations of topological quantum computation, namely the non-Abelian braiding of identical particles, can be physically realized in a general system of Dirac quasiparticles in 1 + 1D. Our method is based on the study of the analytic structure of the different Euclidean correlation functions of Dirac fields, which are conveniently expressed as functions of a complex variable. When the Dirac field is an (Abelian) anyon with statistics parameter s (2s not an integer), we show that the associated Majorana states of such a field present non-Abelian statistics. The explicit form of the unitary, non-commuting (monodromy) matrices generated upon braiding is derived as a function of s and is shown to satisfy the Yang–Baxter algebra. For the special case of s = 1/4, we show that the braiding matrices become the logic gates NOT, CNOT,… required in the algorithms of universal quantum computation. We suggest that maybe polyacetylene, alternately doped with alkali and halogen atoms, is a potential candidate for a physical material realization of the system studied here.

Qubit Approach to the Topological Degeneracy of 4 and 6-Quasihole Pfaffian Paired Hall State

By virtue of the two-layer picture of Pfaffian pair Hall state, a qubit representation of topological degeneracy for quasiholes excitation is displayed. The non-Abelian feature of states can be manifested readily by the new wave functions. The virtue of this approach is that one does not need to find the equalities of Pfaffians, which is a so tedious task as exemplified for the case of six quasiholes. Then the braiding matrices are also constructed readily by just permutating single-qubit states, which are unitary and hermite.