2,289 research outputs found
Using level-2 fuzzy sets to combine uncertainty and imprecision in fuzzy regions
In many applications, spatial data need to be considered but are prone to uncertainty or imprecision. A fuzzy region - a fuzzy set over a two dimensional domain - allows the representation of such imperfect spatial data. In the original model, points of the fuzzy region where treated independently, making it impossible to model regions where groups of points should be considered as one basic element or subregion. A first extension overcame this, but required points within a group to have the same membership grade. In this contribution, we will extend this further, allowing a fuzzy region to contain subregions in which not all points have the same membership grades. The concept can be used as an underlying model in spatial applications, e.g. websites showing maps and requiring representation of imprecise features or websites with routing functions needing to handle concepts as walking distance or closeby
Renormalization and tensor product states in spin chains and lattices
We review different descriptions of many--body quantum systems in terms of
tensor product states. We introduce several families of such states in terms of
known renormalization procedures, and show that they naturally arise in that
context. We concentrate on Matrix Product States, Tree Tensor States,
Multiscale Entanglement Renormalization Ansatz, and Projected Entangled Pair
States. We highlight some of their properties, and show how they can be used to
describe a variety of systems.Comment: Review paper for the special issue of J. Phys.
Continuous Matrix Product States for Quantum Fields
We define matrix product states in the continuum limit, without any reference
to an underlying lattice parameter. This allows to extend the density matrix
renormalization group and variational matrix product state formalism to quantum
field theories and continuum models in 1 spatial dimension. We illustrate our
procedure with the Lieb-Liniger model
Matrix product states represent ground states faithfully
We quantify how well matrix product states approximate exact ground states of
1-D quantum spin systems as a function of the number of spins and the entropy
of blocks of spins. We also investigate the convex set of local reduced density
operators of translational invariant systems. The results give a theoretical
justification for the high accuracy of renormalization group algorithms, and
justifies their use even in the case of critical systems
Mapping local Hamiltonians of fermions to local Hamiltonians of spins
We show how to map local fermionic problems onto local spin problems on a
lattice in any dimension. The main idea is to introduce auxiliary degrees of
freedom, represented by Majorana fermions, which allow us to extend the
Jordan-Wigner transformation to dimensions higher than one. We also discuss the
implications of our results in the numerical investigation of fermionic
systems.Comment: Added explicit mappin
General Monogamy Inequality for Bipartite Qubit Entanglement
We consider multipartite states of qubits and prove that their bipartite
quantum entanglement, as quantified by the concurrence, satisfies a monogamy
inequality conjectured by Coffman, Kundu, and Wootters. We relate this monogamy
inequality to the concept of frustration of correlations in quantum spin
systems.Comment: Fixed spelling mistake. Added references. Fixed error in
transformation law. Shorter and more explicit proof of capacity formula.
Reference added. Rewritten introduction and conclusion
Post-Matrix Product State Methods: To tangent space and beyond
We develop in full detail the formalism of tangent states to the manifold of
matrix product states, and show how they naturally appear in studying
time-evolution, excitations and spectral functions. We focus on the case of
systems with translation invariance in the thermodynamic limit, where momentum
is a well defined quantum number. We present some new illustrative results and
discuss analogous constructions for other variational classes. We also discuss
generalizations and extensions beyond the tangent space, and give a general
outlook towards post matrix product methods.Comment: 40 pages, 8 figure
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