4,023 research outputs found
Kauffman Knot Invariant from SO(N) or Sp(N) Chern-Simons theory and the Potts Model
The expectation value of Wilson loop operators in three-dimensional SO(N)
Chern-Simons gauge theory gives a known knot invariant: the Kauffman
polynomial. Here this result is derived, at the first order, via a simple
variational method. With the same procedure the skein relation for Sp(N) are
also obtained. Jones polynomial arises as special cases: Sp(2), SO(-2) and
SL(2,R). These results are confirmed and extended up to the second order, by
means of perturbation theory, which moreover let us establish a duality
relation between SO(+/-N) and Sp(-/+N) invariants. A correspondence between the
firsts orders in perturbation theory of SO(-2), Sp(2) or SU(2) Chern-Simons
quantum holonomies and the partition function of the Q=4 Potts Model is built.Comment: 20 pages, 7 figures; accepted for publication on Phys. Rev.
Teleportation, Braid Group and Temperley--Lieb Algebra
We explore algebraic and topological structures underlying the quantum
teleportation phenomena by applying the braid group and Temperley--Lieb
algebra. We realize the braid teleportation configuration, teleportation
swapping and virtual braid representation in the standard description of the
teleportation. We devise diagrammatic rules for quantum circuits involving
maximally entangled states and apply them to three sorts of descriptions of the
teleportation: the transfer operator, quantum measurements and characteristic
equations, and further propose the Temperley--Lieb algebra under local unitary
transformations to be a mathematical structure underlying the teleportation. We
compare our diagrammatical approach with two known recipes to the quantum
information flow: the teleportation topology and strongly compact closed
category, in order to explain our diagrammatic rules to be a natural
diagrammatic language for the teleportation.Comment: 33 pages, 19 figures, latex. The present article is a short version
of the preprint, quant-ph/0601050, which includes details of calculation,
more topics such as topological diagrammatical operations and entanglement
swapping, and calls the Temperley--Lieb category for the collection of all
the Temperley--Lieb algebra with physical operations like local unitary
transformation
Quantum entanglement: The unitary 8-vertex braid matrix with imaginary rapidity
We study quantum entanglements induced on product states by the action of
8-vertex braid matrices, rendered unitary with purely imaginary spectral
parameters (rapidity). The unitarity is displayed via the "canonical
factorization" of the coefficients of the projectors spanning the basis. This
adds one more new facet to the famous and fascinating features of the 8-vertex
model. The double periodicity and the analytic properties of the elliptic
functions involved lead to a rich structure of the 3-tangle quantifying the
entanglement. We thus explore the complex relationship between topological and
quantum entanglement.Comment: 4 pages in REVTeX format, 2 figure
Quantum logic as superbraids of entangled qubit world lines
Presented is a topological representation of quantum logic that views
entangled qubit spacetime histories (or qubit world lines) as a generalized
braid, referred to as a superbraid. The crossing of world lines is purely
quantum in nature, most conveniently expressed analytically with
ladder-operator-based quantum gates. At a crossing, independent world lines can
become entangled. Complicated superbraids are systematically reduced by
recursively applying novel quantum skein relations. If the superbraid is closed
(e.g. representing quantum circuits with closed-loop feedback, quantum lattice
gas algorithms, loop or vacuum diagrams in quantum field theory), then one can
decompose the resulting superlink into an entangled superposition of classical
links. In turn, for each member link, one can compute a link invariant, e.g.
the Jones polynomial. Thus, a superlink possesses a unique link invariant
expressed as an entangled superposition of classical link invariants.Comment: 4 page
Graph Invariants of Vassiliev Type and Application to 4D Quantum Gravity
We consider a special class of Kauffman's graph invariants of rigid vertex
isotopy (graph invariants of Vassiliev type). They are given by a functor from
a category of colored and oriented graphs embedded into a 3-space to a category
of representations of the quasi-triangular ribbon Hopf algebra . Coefficients in expansions of them with respect to () are
known as the Vassiliev invariants of finite type. In the present paper, we
construct two types of tangle operators of vertices. One of them corresponds to
a Casimir operator insertion at a transverse double point of Wilson loops. This
paper proposes a non-perturbative generalization of Kauffman's recent result
based on a perturbative analysis of the Chern-Simons quantum field theory. As a
result, a quantum group analog of Penrose's spin network is established taking
into account of the orientation. We also deal with the 4-dimensional canonical
quantum gravity of Ashtekar. It is verified that the graph invariants of
Vassiliev type are compatible with constraints of the quantum gravity in the
loop space representation of Rovelli and Smolin.Comment: 34 pages, AMS-LaTeX, no figures,The proof of thm.5.1 has been
improve
Knots in interaction
We study the geometry of interacting knotted solitons. The interaction is
local and advances either as a three-body or as a four-body process, depending
on the relative orientation and a degeneracy of the solitons involved. The
splitting and adjoining is governed by a four-point vertex in combination with
duality transformations. The total linking number is preserved during the
interaction. It receives contributions both from the twist and the writhe,
which are variable. Therefore solitons can twine and coil and links can be
formed.Comment: figures now in GIF forma
Lens Spaces and Handlebodies in 3D Quantum Gravity
We calculate partition functions for lens spaces L_{p,q} up to p=8 and for
genus 1 and 2 handlebodies H_1, H_2 in the Turaev-Viro framework. These can be
interpreted as transition amplitudes in 3D quantum gravity. In the case of lens
spaces L_{p,q} these are vacuum-to-vacuum amplitudes \O -> \O, whereas for
the 1- and 2-handlebodies H_1, H_2 they represent genuinely topological
transition amplitudes \O -> T^2 and \O -> T^2 # T^2, respectively.Comment: 14 pages, LaTeX, 5 figures, uses eps
Canonical quantum gravity in the Vassiliev invariants arena: I. Kinematical structure
We generalize the idea of Vassiliev invariants to the spin network context,
with the aim of using these invariants as a kinematical arena for a canonical
quantization of gravity. This paper presents a detailed construction of these
invariants (both ambient and regular isotopic) requiring a significant
elaboration based on the use of Chern-Simons perturbation theory which extends
the work of Kauffman, Martin and Witten to four-valent networks. We show that
this space of knot invariants has the crucial property -from the point of view
of the quantization of gravity- of being loop differentiable in the sense of
distributions. This allows the definition of diffeomorphism and Hamiltonian
constraints. We show that the invariants are annihilated by the diffeomorphism
constraint. In a companion paper we elaborate on the definition of a
Hamiltonian constraint, discuss the constraint algebra, and show that the
construction leads to a consistent theory of canonical quantum gravity.Comment: 21 Pages, RevTex, many figures included with psfi
Experimental approximation of the Jones polynomial with DQC1
We present experimental results approximating the Jones polynomial using 4
qubits in a liquid state nuclear magnetic resonance quantum information
processor. This is the first experimental implementation of a complete problem
for the deterministic quantum computation with one quantum bit model of quantum
computation, which uses a single qubit accompanied by a register of completely
random states. The Jones polynomial is a knot invariant that is important not
only to knot theory, but also to statistical mechanics and quantum field
theory. The implemented algorithm is a modification of the algorithm developed
by Shor and Jordan suitable for implementation in NMR. These experimental
results show that for the restricted case of knots whose braid representations
have four strands and exactly three crossings, identifying distinct knots is
possible 91% of the time.Comment: 5 figures. Version 2 changes: published version, minor errors
corrected, slight changes to improve readabilit
The Asymptotic Number of Attractors in the Random Map Model
The random map model is a deterministic dynamical system in a finite phase
space with n points. The map that establishes the dynamics of the system is
constructed by randomly choosing, for every point, another one as being its
image. We derive here explicit formulas for the statistical distribution of the
number of attractors in the system. As in related results, the number of
operations involved by our formulas increases exponentially with n; therefore,
they are not directly applicable to study the behavior of systems where n is
large. However, our formulas lend themselves to derive useful asymptotic
expressions, as we show.Comment: 16 pages, 1 figure. Minor changes. To be published in Journal of
Physics A: Mathematical and Genera
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