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Title: Entropy and Exact Matrix-Product Representation of the Laughlin Wave Function

Abstract

An analytical expression for the von Neumann entropy of the Laughlin wave function is obtained for any possible bipartition between the particles described by this wave function, for a filling fraction {nu}=1. Also, for a filling fraction {nu}=1/m, where m is an odd integer, an upper bound on this entropy is exhibited. These results yield a bound on the smallest possible size of the matrices for an exact representation of the Laughlin ansatz in terms of a matrix-product state. An analytical matrix-product state representation of this state is proposed in terms of representations of the Clifford algebra. For {nu}=1, this representation is shown to be asymptotically optimal in the limit of a large number of particles.

Authors:
;  [1];  [1];  [2]
  1. Departament d'Estructura i Constituents de la Materia, Universitat de Barcelona, 647 Diagonal, 08028 Barcelona (Spain)
  2. (Australia)
Publication Date:
OSTI Identifier:
20955433
Resource Type:
Journal Article
Resource Relation:
Journal Name: Physical Review Letters; Journal Volume: 98; Journal Issue: 6; Other Information: DOI: 10.1103/PhysRevLett.98.060402; (c) 2007 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CLIFFORD ALGEBRA; ENTROPY; EXACT SOLUTIONS; MATRICES; WAVE FUNCTIONS

Citation Formats

Iblisdir, S., Latorre, J. I., Orus, R., and School of Physical Sciences, University of Queensland, QLD 4072. Entropy and Exact Matrix-Product Representation of the Laughlin Wave Function. United States: N. p., 2007. Web. doi:10.1103/PHYSREVLETT.98.060402.
Iblisdir, S., Latorre, J. I., Orus, R., & School of Physical Sciences, University of Queensland, QLD 4072. Entropy and Exact Matrix-Product Representation of the Laughlin Wave Function. United States. doi:10.1103/PHYSREVLETT.98.060402.
Iblisdir, S., Latorre, J. I., Orus, R., and School of Physical Sciences, University of Queensland, QLD 4072. Fri . "Entropy and Exact Matrix-Product Representation of the Laughlin Wave Function". United States. doi:10.1103/PHYSREVLETT.98.060402.
@article{osti_20955433,
title = {Entropy and Exact Matrix-Product Representation of the Laughlin Wave Function},
author = {Iblisdir, S. and Latorre, J. I. and Orus, R. and School of Physical Sciences, University of Queensland, QLD 4072},
abstractNote = {An analytical expression for the von Neumann entropy of the Laughlin wave function is obtained for any possible bipartition between the particles described by this wave function, for a filling fraction {nu}=1. Also, for a filling fraction {nu}=1/m, where m is an odd integer, an upper bound on this entropy is exhibited. These results yield a bound on the smallest possible size of the matrices for an exact representation of the Laughlin ansatz in terms of a matrix-product state. An analytical matrix-product state representation of this state is proposed in terms of representations of the Clifford algebra. For {nu}=1, this representation is shown to be asymptotically optimal in the limit of a large number of particles.},
doi = {10.1103/PHYSREVLETT.98.060402},
journal = {Physical Review Letters},
number = 6,
volume = 98,
place = {United States},
year = {Fri Feb 09 00:00:00 EST 2007},
month = {Fri Feb 09 00:00:00 EST 2007}
}
  • A coherent, intrinsic, basis-set-independent analysis is developed for the invariants of the first-order density matrix of an accurate molecular electronic wavefunction. From the hierarchical ordering of the natural orbitals, the zeroth-order orbital space is deduced, which generates the zeroth-order wavefunction, typically an MCSCF function in the full valence space. It is shown that intrinsically embedded in such wavefunctions are elements that are local in bond regions and elements that are local in atomic regions. Basis-set-independent methods are given that extract and exhibit the intrinsic bond orbitals and the intrinsic minimal-basis quasi-atomic orbitals in terms of which the wavefunction can bemore » exactly constructed. The quasi-atomic orbitals are furthermore oriented by a basis-set independent method (viz. maximization of the sum of the fourth powers of all off-diagonal density matrix elements) so as to exhibit clearly the chemical interactions. The unbiased nature of the method allows for the adaptation of the localized and directed orbitals to changing geometries.« less
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  • We show that the introduction of a more general closed-shell operator allows one to extend Laughlin{close_quote}s wave function to account for the richer hierarchies (1/3,2/5,3/7...;1/5, 2/9,3/13,...,etc.) found experimentally. The construction identifies the special hierarchy states with condensates of correlated electron clusters. This clustering implies a single-particle algebra within the first Landau level (LL) identical to that of multiply filled LLs in the integer quantum Hall effect. The end result is a simple generalized wave function that reproduces the results of both Laughlin and Jain, without reference to higher LLs or projection. {copyright} {ital 1996 The American Physical Society.}
  • We construct a quantum algorithm that creates the Laughlin state for an arbitrary number n of particles for a filling fraction of one. This quantum circuit is efficient since it only uses n(n-1)/2 local qudit gates and its depth scales as 2n-3. Furthermore, we prove the optimality of the circuit using permutation theory arguments and we compute exactly how entanglement develops along the action of each gate. Finally, we discuss its experimental feasibility by decomposing the qudits and the gates in terms of qubits and two qubit-gates, as well as its generalization to arbitrary filling fraction.