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A FINITENESS PROPERTY FOR BRAIDED FUSION CATEGORIES
"... Abstract. We introduce a finiteness property for braided fusion categories, describe a conjecture that would characterize categories possessing this, and verify the conjecture in a number of important cases. In particular we say a category has property F if the associated braid group representations ..."
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Abstract. We introduce a finiteness property for braided fusion categories, describe a conjecture that would characterize categories possessing this, and verify the conjecture in a number of important cases. In particular we say a category has property F if the associated braid group representations factor over a finite group, and suggest that categories of integral FrobeniusPerron dimension are precisely those with property F. 1.
Braid Group and Temperley–Lieb Algebra, and Quantum . . .
, 2008
"... In this paper, we explore algebraic structures and low dimensional topology underlying quantum information and computation. We revisit quantum teleportation from the perspective of the braid group, the symmetric group and the virtual braid group, and propose the braid teleportation, the teleportatio ..."
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In this paper, we explore algebraic structures and low dimensional topology underlying quantum information and computation. We revisit quantum teleportation from the perspective of the braid group, the symmetric group and the virtual braid group, and propose the braid teleportation, the teleportation swapping and the virtual braid teleportation, respectively. Besides, we present a physical interpretation for the braid teleportation and explain it as a sort of crossed measurement. On the other hand, we propose the extended Temperley–Lieb diagrammatical approach to various topics including quantum teleportation, entanglement swapping, universal quantum computation, quantum information flow, and etc. The extended Temperley–Lieb diagrammatical rules are devised to present a diagrammatical representation for the extended Temperley–Lieb category which is the collection of all the Temperley–Lieb algebras with local unitary transformations. In this approach, various descriptions of quantum teleportation are unified in a diagrammatical sense, universal quantum computation is performed with the help of topologicallike features, and quantum information flow is
BRAID REPRESENTATIONS FROM QUANTUM GROUPS OF EXCEPTIONAL LIE TYPE
"... Abstract. We study the problem of determining if the braid group representations obtained from quantum groups of types E, F and G at roots of unity have infinite image or not. In particular we show that when the fusion categories associated with these quantum groups are not weakly integral, the brai ..."
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Abstract. We study the problem of determining if the braid group representations obtained from quantum groups of types E, F and G at roots of unity have infinite image or not. In particular we show that when the fusion categories associated with these quantum groups are not weakly integral, the braid group images are infinite. This provides further evidence for a recent conjecture that weak integrality is necessary and sufficient for the braid group representations associated with any braided fusion category to have finite image. 1.
Finite linear quotients of B3 of low dimension
"... Abstract. We study the problem of deciding whether or not the image of an irreducible representation of the braid group B3 of degree ≤ 5 has finite image if we are only given the eigenvalues of a generator. We provide a partial algorithm that determines when the images are finite or infinite in all ..."
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Abstract. We study the problem of deciding whether or not the image of an irreducible representation of the braid group B3 of degree ≤ 5 has finite image if we are only given the eigenvalues of a generator. We provide a partial algorithm that determines when the images are finite or infinite in all but finitely many cases, and use these results to study examples coming from quantum groups. Our technique uses two classification theorems and the computational group theory package GAP. 1.
BRAID GROUP REPRESENTATIONS FROM TWISTED QUANTUM DOUBLES OF FINITE GROUPS
, 2007
"... Abstract. We investigate the braid group representations arising from categories of representations of twisted quantum doubles of finite groups. For these categories, we show that the resulting braid group representations always factor through finite groups, in contrast to the categories associated ..."
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Abstract. We investigate the braid group representations arising from categories of representations of twisted quantum doubles of finite groups. For these categories, we show that the resulting braid group representations always factor through finite groups, in contrast to the categories associated with quantum groups at roots of unity. We also show that in the case of pgroups, the corresponding pure braid group representations factor through a finite pgroup, which answers a question asked of the first author by V. Drinfeld. 1.
LOCALIZATION OF UNITARY BRAID GROUP REPRESENTATIONS
"... Abstract. Governed by locality, we explore a connection between unitary braid group representations associated to a unitary Rmatrix and to a simple object in a unitary braided fusion category. Unitary Rmatrices, namely unitary solutions to the YangBaxter equation, afford explicitly local unitary ..."
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Abstract. Governed by locality, we explore a connection between unitary braid group representations associated to a unitary Rmatrix and to a simple object in a unitary braided fusion category. Unitary Rmatrices, namely unitary solutions to the YangBaxter equation, afford explicitly local unitary representations of braid groups. Inspired by topological quantum computation, we study whether or not it is possible to reassemble the irreducible summands appearing in the unitary braid group representations from a unitary braided fusion category with possibly different positive multiplicities to get representations that are uniformly equivalent to the ones from a unitary Rmatrix. Such an equivalence will be called a localization of the unitary braid group representations. We show that the q = e πi/6 specialization of the unitary Jones representation of the braid groups can be localized by a unitary 9 × 9 Rmatrix. Actually this Jones representation is the first one in a family of theories (SO(N), 2) for an odd prime N> 1, which are conjectured to be localizable. We formulate several general conjectures and discuss possible connections to physics and computer science. 1.
Teleportation, Braid Group and Temperley–Lieb Algebra”, quantph/0601050
 23 George Svetlichny, Foundations of Physics
, 1981
"... ..."
Quantum Error Correction Code in the Hamiltonian Formulation
, 2008
"... The Hamiltonian model of quantum error correction code in the literature is often constructed with the help of its stabilizer formalism. But there have been many known examples of nonadditive codes which are beyond the standard quantum error correction theory using the stabilizer formalism. In this ..."
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The Hamiltonian model of quantum error correction code in the literature is often constructed with the help of its stabilizer formalism. But there have been many known examples of nonadditive codes which are beyond the standard quantum error correction theory using the stabilizer formalism. In this paper, we suggest the other type of Hamiltonian formalism for quantum error correction code without involving the stabilizer formalism, and explain it by studying the Shor ninequbit code and its generalization. In this Hamiltonian formulation, the unitary evolution operator at a specific time is a unitary basis transformation matrix from the product basis to the quantum error correction code. This basis transformation matrix acts as an entangling quantum operator transforming a separate state to an entangled one, and hence the entanglement nature of the quantum error correction code can be explicitly shown up. Furthermore, as it forms a unitary representation of the Artin braid group, the quantum error correction code can be described by a braiding operator. Moreover, as the unitary evolution operator is a solution of the quantum Yang–Baxter equation, the corresponding Hamiltonian model can be explained as an integrable model in the Yang–Baxter theory. On the other hand, we generalize the Shor ninequbit code and articulate a topic called quantum error correction codes using GreenbergerHorneZeilinger states to yield new nonadditive codes and channeladapted codes.
GENERALIZED AND QUASILOCALIZATIONS OF BRAID GROUP REPRESENTATIONS
"... Abstract. We develop a theory of localization for braid group representations associatedwithobjectsinbraidedfusioncategoriesand,moregenerally,toYangBaxter operators in monoidal categories. The essential problem is to determine when a family of braid representations can be uniformly modelled upon a ..."
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Abstract. We develop a theory of localization for braid group representations associatedwithobjectsinbraidedfusioncategoriesand,moregenerally,toYangBaxter operators in monoidal categories. The essential problem is to determine when a family of braid representations can be uniformly modelled upon a tensor power of a fixed vector space in such a way that the braid group generators act “locally”. Although related to the notion of (quasi)fiber functors for fusion categories,remarkably,suchlocalizationscanexistforrepresentationsassociated with objects of nonintegral dimension. We conjecture that such localizations exist precisely when the object in question has dimension the squareroot of an integer and prove several key special cases of the conjecture. 1.
Laboratoire de Physique Quantique de la Matière et de Modélisations Mathématiques,
, 2007
"... We construct (2n) 2 × (2n) 2 unitary braid matrices ̂ R for n ≥ 2 generalizing the class known for n = 1. A set of (2n) × (2n) matrices (I,J,K,L) are defined. ̂R is expressed in terms of their tensor products (such as K ⊗ J), leading to a canonical formulation for all n. Complex projectors P ± prov ..."
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We construct (2n) 2 × (2n) 2 unitary braid matrices ̂ R for n ≥ 2 generalizing the class known for n = 1. A set of (2n) × (2n) matrices (I,J,K,L) are defined. ̂R is expressed in terms of their tensor products (such as K ⊗ J), leading to a canonical formulation for all n. Complex projectors P ± provide a basis for our real, unitary ̂ R. Baxterization is obtained. Diagonalizations and blockdiagonalizations are presented. The loss of braid property when ̂ R (n> 1) is blockdiagonalized in terms of ̂ R (n = 1) is pointed out and explained. For odd dimension (2n + 1) 2 × (2n + 1) 2, a previously constructed braid matrix is complexified to obtain unitarity. ̂ RLL and ̂ RTTalgebras, chain Hamiltonians, potentials for factorizable Smatrices, complex noncommutative spaces are all studied briefly in the context of our unitary braid matrices. Turaev construction of link invariants is formulated for our case. We conclude with comments concerning entanglements. 1