3,411 research outputs found
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
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
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
An extended Hubbard model with ring exchange: a route to a non-Abelian topological phase
We propose an extended Hubbard model on a 2D Kagome lattice with an
additional ring-exchange term. The particles can be either bosons or spinless
fermions . At a special filling fraction of 1/6 the model is analyzed in the
lowest non-vanishing order of perturbation theory. Such ``undoped'' model is
closely related to the Quantum Dimer Model. We show how to arrive at an exactly
soluble point whose ground state manifold is the extensively degenerate
``d-isotopy space'', a necessary precondition for for a certain type of
non-Abelian topological order. Near the ``special'' values, , this space is expected to collapse to a stable topological phase
with anyonic excitations closely related to SU(2) Chern-Simons theory at level
k.Comment: 4 pages, 2 colour figures, submitted to PRL. For an extended
treatment of a more general family of models see cond-mat/030912
On the Quantum Computational Complexity of the Ising Spin Glass Partition Function and of Knot Invariants
It is shown that the canonical problem of classical statistical
thermodynamics, the computation of the partition function, is in the case of
+/-J Ising spin glasses a particular instance of certain simple sums known as
quadratically signed weight enumerators (QWGTs). On the other hand it is known
that quantum computing is polynomially equivalent to classical probabilistic
computing with an oracle for estimating QWGTs. This suggests a connection
between the partition function estimation problem for spin glasses and quantum
computation. This connection extends to knots and graph theory via the
equivalence of the Kauffman polynomial and the partition function for the Potts
model.Comment: 8 pages, incl. 2 figures. v2: Substantially rewritte
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