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2010

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This paper discusses relationships between topological entangle-ment and quantum entanglement. Specifically, we propose that it is more fundamental to view topological entanglements such as braids as entanglement operators and to associate with them unitary operators that are capable of creating quantum entanglement.

The purpose of these lecture notes is to provide readers, who have some mathematical background but little or no exposure to quantum mechanics and quantum computation, with enough material to begin reading the research literature in quantum computation, quantum cryptography, and quantum information theory. This paper is a written version of the first of… (More)

In this paper, we give a description of a recent quantum algorithm created by Aharonov, Jones, and Landau for approximating the values of the Jones polynomial at roots of unity of the form e 2πi/k. This description is given with two objectives in mind. The first is to describe the algorithm in such a way as to make explicit the underlying and inherent… (More)

This paper gives a criterion for detecting the entanglement of a quantum state, and uses it to study the relationship between topological and quantum entanglement. It is fundamental to view topological entanglements such as braids as entanglement operators and to associate to them unitary operators that are capable of creating quantum entanglement. The… (More)

We review the q-deformed spin network approach to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups.

- S. J. LOMONACO, R. H. Fox
- 1981

Let K be a CW complex with an aspherical splitting, i.e., with subcomplexes ϋΓ_ and K + such that (a) K—K-UK+ and (b) iΓ_, K 0 =K-ΠK +1 K + are connected and aspherical. The main theorem of this paper gives a practical procedure for computing the homology H*K of the universal cover K of K. It also provides a practical method for computing the algebraic… (More)

This paper proposes the definition of a quantum knot as a linear superposition of classical knots in three dimensional space. The definition is constructed and applications are discussed. Then the paper details extensions and also limitations of the Aravind Hypothesis for comparing quantum measurement with classical topological measurement. We propose a… (More)

This paper is dedicated to new progress in the relationship of topology and quantum physics. Abstract. This paper is an introduction to relationships between quantum topology and quantum computing. In this paper we discuss unitary solutions to the Yang-Baxter equation that are universal quantum gates, quantum en-tanglement and topological entanglement, and… (More)

We give an elementary construction of the Fibonacci model, a unitary braid group representation that is universal for quantum computation. This paper is dedicated to Professor C. N. Yang, on his 85-th birthday.