119,087 research outputs found
Characterization of equivariant maps and application to entanglement detection
We study equivariant linear maps between finite-dimensional matrix algebras,
as introduced by Bhat. These maps satisfy an algebraic property which makes it
easy to study their positivity or k-positivity. They are therefore particularly
suitable for applications to entanglement detection in quantum information
theory. We characterize their Choi matrices. In particular, we focus on a
subfamily that we call (a, b)-unitarily equivariant. They can be seen as both a
generalization of maps invariant under unitary conjugation as studied by Bhat
and as a generalization of the equivariant maps studied by Collins et al. Using
representation theory, we fully compute them and study their graphical
representation, and show that they are basically enough to study all
equivariant maps. We finally apply them to the problem of entanglement
detection and prove that they form a sufficient (infinite) family of positive
maps to detect all k-entangled density matrices.Comment: 16 pages, 4 figure
Generalized trace and modified dimension functions on ribbon categories
In this paper we use topological techniques to construct generalized trace
and modified dimension functions on ideals in certain ribbon categories.
Examples of such ribbon categories naturally arise in representation theory
where the usual trace and dimension functions are zero, but these generalized
trace and modified dimension functions are non-zero. Such examples include
categories of finite dimensional modules of certain Lie algebras and finite
groups over a field of positive characteristic and categories of finite
dimensional modules of basic Lie superalgebras over the complex numbers. These
modified dimensions can be interpreted categorically and are closely related to
some basic notions from representation theory.Comment: 44 page
Quasi-Quantum Groups, Knots, Three-Manifolds, and Topological Field Theory
We show how to construct, starting from a quasi-Hopf algebra, or
quasi-quantum group, invariants of knots and links. In some cases, these
invariants give rise to invariants of the three-manifolds obtained by surgery
along these links. This happens for a finite-dimensional quasi-quantum group,
whose definition involves a finite group , and a 3-cocycle \om, which was
first studied by Dijkgraaf, Pasquier and Roche. We treat this example in more
detail, and argue that in this case the invariants agree with the partition
function of the topological field theory of Dijkgraaf and Witten depending on
the same data G, \,\om.Comment: 30 page
Tensor network states and algorithms in the presence of a global U(1) symmetry
Tensor network decompositions offer an efficient description of certain
many-body states of a lattice system and are the basis of a wealth of numerical
simulation algorithms. In a recent paper [arXiv:0907.2994v1] we discussed how
to incorporate a global internal symmetry, given by a compact, completely
reducible group G, into tensor network decompositions and algorithms. Here we
specialize to the case of Abelian groups and, for concreteness, to a U(1)
symmetry, often associated with particle number conservation. We consider
tensor networks made of tensors that are invariant (or covariant) under the
symmetry, and explain how to decompose and manipulate such tensors in order to
exploit their symmetry. In numerical calculations, the use of U(1) symmetric
tensors allows selection of a specific number of particles, ensures the exact
preservation of particle number, and significantly reduces computational costs.
We illustrate all these points in the context of the multi-scale entanglement
renormalization ansatz.Comment: 22 pages, 25 figures, RevTeX
NITELIGHT: A Graphical Tool for Semantic Query Construction
Query formulation is a key aspect of information retrieval, contributing to both the efficiency and usability of many semantic applications. A number of query languages, such as SPARQL, have been developed for the Semantic Web; however, there are, as yet, few tools to support end users with respect to the creation and editing of semantic queries. In this paper we introduce a graphical tool for semantic query construction (NITELIGHT) that is based on the SPARQL query language specification. The tool supports end users by providing a set of graphical notations that represent semantic query language constructs. This language provides a visual query language counterpart to SPARQL that we call vSPARQL. NITELIGHT also provides an interactive graphical editing environment that combines ontology navigation capabilities with graphical query visualization techniques. This paper describes the functionality and user interaction features of the NITELIGHT tool based on our work to date. We also present details of the vSPARQL constructs used to support the graphical representation of SPARQL queries
Chiral Ring of Strange Metals: The Multicolor Limit
The low energy limit of a dense 2D adjoint QCD is described by a family of
supersymmetric coset conformal field theories. In previous
work we constructed chiral primaries for a small number of colors. Our
aim in the present note is to determine the chiral ring in the multicolor limit
where is sent to infinity. We shall find that chiral primaries are labeled
by partitions and identify the ring they generate as the ring of Schur
polynomials. Our findings impose strong constraints on the possible dual
description through string theory in an compactification.Comment: 25 pages, 4 figure
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