4,451 research outputs found
Teleportation Topology
We discuss the structure of teleportation. By associating matrices to the
preparation and measurement states, we show that for a unitary transformation M
there is a full teleportation procedure for obtaining M|S> from a given state
|S>. The key to this construction is a diagrammatic intepretation of matrix
multiplication that applies equally well to a topological composition of a
maximum and a minimum that underlies the structure of the teleportation. This
paper is a preliminary report on joint work with H. Carteret and S. Lomonaco.Comment: LaTeX document, 16 pages, 8 figures, Talk delivered at the Xth
International Conference on Quantum Optics, Minsk, Belaru
Quantum entanglement, unitary braid representation and Temperley-Lieb algebra
Important developments in fault-tolerant quantum computation using the
braiding of anyons have placed the theory of braid groups at the very
foundation of topological quantum computing. Furthermore, the realization by
Kauffman and Lomonaco that a specific braiding operator from the solution of
the Yang-Baxter equation, namely the Bell matrix, is universal implies that in
principle all quantum gates can be constructed from braiding operators together
with single qubit gates. In this paper we present a new class of braiding
operators from the Temperley-Lieb algebra that generalizes the Bell matrix to
multi-qubit systems, thus unifying the Hadamard and Bell matrices within the
same framework. Unlike previous braiding operators, these new operators
generate {\it directly}, from separable basis states, important entangled
states such as the generalized Greenberger-Horne-Zeilinger states, cluster-like
states, and other states with varying degrees of entanglement.Comment: 5 pages, no figur
On homotopies with triple points of classical knots
We consider a knot homotopy as a cylinder in 4-space. An ordinary triple
point of the cylinder is called {\em coherent} if all three branches
intersect at pairwise with the same index. A {\em triple unknotting} of a
classical knot is a homotopy which connects with the trivial knot and
which has as singularities only coherent triple points. We give a new formula
for the first Vassiliev invariant by using triple unknottings. As a
corollary we obtain a very simple proof of the fact that passing a coherent
triple point always changes the knot type. As another corollary we show that
there are triple unknottings which are not homotopic as triple unknottings even
if we allow more complicated singularities to appear in the homotopy of the
homotopy.Comment: 10 pages, 13 figures, bugs in figures correcte
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
On a suggestion relating topological and quantum mechanical entanglements
We analyze a recent suggestion \cite{kauffman1,kauffman2} on a possible
relation between topological and quantum mechanical entanglements. We show that
a one to one correspondence does not exist, neither between topologically
linked diagrams and entangled states, nor between braid operators and quantum
entanglers. We also add a new dimension to the question of entangling
properties of unitary operators in general.Comment: RevTex, 7 eps figures, to be published in Phys. Lett. A (2004
The Politics of Rights of Nature
How Rights of Nature laws are transforming governance to address environmental crises through more ecologically sustainable approaches to development. With the window of opportunity to take meaningful action on climate change and mass extinction closing, a growing number of communities, organizations, and governments around the world are calling for Rights of Nature (RoN) to be legally recognized. RoN advocates are creating new laws that recognize natural ecosystems as subjects with inherent rights, and appealing to courts to protect those rights. Going beyond theory and philosophy, in this book Craig Kauffman and Pamela Martin analyze the politics behind the creation and implementation of these laws, as well as the effects of the laws on the politics of sustainable development. Kauffman and Martin tell how community activists, lawyers, judges, scientists, government leaders, and ordinary citizens have formed a global movement to advance RoN as a solution to the environmental crises facing the planet. They compare successful and failed attempts to implement RoN at various levels of government in six countries—Bolivia, Colombia, Ecuador, India, New Zealand, and the United States—asking why these laws emerged and proliferated in the mid-2000s, why they construct RoN differently, and why some efforts at implementation are more successful than others. As they analyze efforts to use RoN as a tool for constructing more ecocentric sustainable development, capable of achieving the 2030 Agenda for Sustainable Development goal of living “in harmony with Nature,” Kauffman and Martin show how RoN jurisprudence evolves through experimentation and reshapes the debates surrounding sustainable development
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
An analytic Approach to Turaev's Shadow Invariant
In the present paper we extend the "torus gauge fixing approach" by Blau and
Thompson (Nucl. Phys. B408(1):345--390, 1993) for Chern-Simons models with base
manifolds M of the form M= \Sigma x S^1 in a suitable way. We arrive at a
heuristic path integral formula for the Wilson loop observables associated to
general links in M. We then show that the right-hand side of this formula can
be evaluated explicitly in a non-perturbative way and that this evaluation
naturally leads to the face models in terms of which Turaev's shadow invariant
is defined.Comment: 44 pages, 2 figures. Changes have been made in Sec. 2.3, Sec 2.4,
Sec. 3.4, and Sec. 3.5. Appendix C is ne
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