Advances in Quantum Computation: Representation Theory, by Kazem Mahdavi, Deborah Koslover

By Kazem Mahdavi, Deborah Koslover

This quantity represents the talks given on the convention on Interactions among illustration thought, Quantum box conception, type conception, Mathematical Physics, and Quantum details concept, held in September 2007 on the college of Texas at Tyler. The papers during this quantity, written via best specialists within the box, handle actual elements, mathematical facets, and foundational problems with quantum computation. This quantity will gain researchers drawn to advances in quantum computation and verbal exchange, in addition to graduate scholars who desire to input the sector of quantum computation

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BRAID GROUP, TEMPERLEY-LIEB ALGEBRA, AND QUANTUM INFORMATION 53 55 T |ψ B Teleportation |φ+ CA : Id |φ− CA : σ3 |ψ + CA : σ1 |ψ − CA : iσ2 Unitary Corrections Classical Channel Bell Measurement State Preparation |ψ ± CA ψ±| |φ± CA φ± | |ψ Charlie |φ+ C AB Alice Bob X Figure 1. Diagrammatical description of quantum teleportation. , embedding the right matrix into the left one. With the product basis |ij , i, j = 0, 1, four mutually orthogonal Bell states have the form, 1 1 |φ+ = √ (|00 + |11 ), |φ− = √ (|00 − |11 ), 2 2 1 1 |ψ + = √ (|01 + |10 ), (5) |ψ − = √ (|01 − |10 ), 2 2 which derive the product basis |ij in terms of Bell states, 1 1 |01 = √ (|ψ + + |ψ − ), |00 = √ (|φ+ + |φ− ), 2 2 1 1 + − (6) |11 = √ (|φ+ − |φ− ).

In a joint article with Kauffman and Werner [31], we make a survey of diagrammatical tensor calculus and matrix representations, and explore topological and algebraic structures underlying multipartite entanglements. In this article as a further extension of our work [32, 33, 34], we describe quantum information and computation (especially quantum teleportation [8, 9]) in the language of the braid group and Temperley–Lieb (TL) algebra [35]. We propose the braid teleportation, teleportation swapping and virtual braid teleportation, and devise the extended TL diagrammatical rules to describe quantum teleportation, entanglement swapping, universal quantum computation and quantum information flow.

S. V. Porto, Lattice of double wells for manipulating pairs of cold atoms, Phys. Rev. A. 73 (2006), 033605. [20] G. Trigg, Quantum Mechanics, D. Van Nostrand Company, Princeton, New Jersey, 1964. edu  ? >D5 = @ ? >CD1>D D? >  1D? >4C D? @6 1<752B1 /0 B59=5B 1D ""! >41BI  &   )     "    ( (  % $ ! B G9C85C D? > D? ! 6 )5H1C 1D )I<5B 3 American c  5 >D5 B > 1 =Mathematical 5 ? 6 3 ? @I B97 8DSociety 8 ? > B59=5BNC =1@ @9>7 1>4 1D? > 7B1@8C G9D8 DG?

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