46 research outputs found
Miscellaneous Applications of Quons
This paper deals with quon algebras or deformed oscillator algebras, for
which the deformation parameter is a root of unity. We show the interest of
such algebras for fractional supersymmetric quantum mechanics, angular momentum
theory and quantum information. More precisely, quon algebras are used for (i)
a realization of a generalized Weyl-Heisenberg algebra from which it is
possible to associate a fractional supersymmetric dynamical system, (ii) a
polar decomposition of SU_2 and (iii) a construction of mutually unbiased bases
in Hilbert spaces of prime dimension. We also briefly discuss (symmetric
informationally complete) positive operator valued measures in the spirit of
(iii).Comment: This is a contribution to the Proc. of the 3-rd Microconference
"Analytic and Algebraic Methods III"(June 19, 2007, Prague, Czech Republic),
published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
Unitary reflection groups for quantum fault tolerance
This paper explores the representation of quantum computing in terms of
unitary reflections (unitary transformations that leave invariant a hyperplane
of a vector space). The symmetries of qubit systems are found to be supported
by Euclidean real reflections (i.e., Coxeter groups) or by specific imprimitive
reflection groups, introduced (but not named) in a recent paper [Planat M and
Jorrand Ph 2008, {\it J Phys A: Math Theor} {\bf 41}, 182001]. The
automorphisms of multiple qubit systems are found to relate to some Clifford
operations once the corresponding group of reflections is identified. For a
short list, one may point out the Coxeter systems of type and (for
single qubits), and (for two qubits), and (for three
qubits), the complex reflection groups and groups No 9 and 31 in
the Shephard-Todd list. The relevant fault tolerant subsets of the Clifford
groups (the Bell groups) are generated by the Hadamard gate, the phase
gate and an entangling (braid) gate [Kauffman L H and Lomonaco S J 2004 {\it
New J. of Phys.} {\bf 6}, 134]. Links to the topological view of quantum
computing, the lattice approach and the geometry of smooth cubic surfaces are
discussed.Comment: new version for the Journal of Computational and Theoretical
Nanoscience, focused on "Technology Trends and Theory of Nanoscale Devices
for Quantum Applications
An angular momentum approach to quadratic Fourier transform, Hadamard matrices, Gauss sums, mutually unbiased bases, unitary group and Pauli group
The construction of unitary operator bases in a finite-dimensional Hilbert
space is reviewed through a nonstandard approach combinining angular momentum
theory and representation theory of SU(2). A single formula for the bases is
obtained from a polar decomposition of SU(2) and analysed in terms of cyclic
groups, quadratic Fourier transforms, Hadamard matrices and generalized Gauss
sums. Weyl pairs, generalized Pauli operators and their application to the
unitary group and the Pauli group naturally arise in this approach.Comment: Topical review (40 pages). Dedicated to the memory of Yurii
Fedorovich Smirno
ON TWO WAYS TO LOOK FOR MUTUALLY UNBIASED BASES
Two equivalent ways of looking for mutually unbiased bases are discussed in this note. The passage from the search for d+1 mutually unbiased bases in Cd to the search for d(d+1) vectors in Cd2 satisfying constraint relations is clarified. Symmetric informationally complete positive-operator-valued measures are briefly discussed in a similar vein
Quantum Entanglement and Projective Ring Geometry
The paper explores the basic geometrical properties of the observables
characterizing two-qubit systems by employing a novel projective ring geometric
approach. After introducing the basic facts about quantum complementarity and
maximal quantum entanglement in such systems, we demonstrate that the
1515 multiplication table of the associated four-dimensional matrices
exhibits a so-far-unnoticed geometrical structure that can be regarded as three
pencils of lines in the projective plane of order two. In one of the pencils,
which we call the kernel, the observables on two lines share a base of Bell
states. In the complement of the kernel, the eight vertices/observables are
joined by twelve lines which form the edges of a cube. A substantial part of
the paper is devoted to showing that the nature of this geometry has much to do
with the structure of the projective lines defined over the rings that are the
direct product of copies of the Galois field GF(2), with = 2, 3 and 4.Comment: 13 pages, 6 figures Fig. 3 improved, typos corrected; Version 4:
Final Version Published in SIGMA (Symmetry, Integrability and Geometry:
Methods and Applications) at http://www.emis.de/journals/SIGMA
Generalized spin bases for quantum chemistry and quantum information
Symmetry adapted bases in quantum chemistry and bases adapted to quantum
information share a common characteristics: both of them are constructed from
subspaces of the representation space of the group SO(3) or its double group
(i.e., spinor group) SU(2). We exploit this fact for generating spin bases of
relevance for quantum systems with cyclic symmetry and equally well for quantum
information and quantum computation. Our approach is based on the use of
generalized Pauli matrices arising from a polar decomposition of SU(2). This
approach leads to a complete solution for the construction of mutually unbiased
bases in the case where the dimension d of the considered Hilbert subspace is a
prime number. We also give the starting point for studying the case where d is
the power of a prime number. A connection of this work with the unitary group
U(d) and the Pauli group is brielly underlined.Comment: Dedicated to Professor Rudolf Zahradnik on the occasion of his 80th
birthday. Invited paper to be published in Collection of Czechoslovak
Chemical Communication
Bosonic and k-fermionic coherent states for a class of polynomial Weyl-Heisenberg algebras
The aim of this article is to construct \`a la Perelomov and \`a la
Barut-Girardello coherent states for a polynomial Weyl-Heisenberg algebra. This
generalized Weyl-Heisenberg algebra, noted A(x), depends on r real parameters
and is an extension of the one-parameter algebra introduced in Daoud M and
Kibler MR 2010 J. Phys. A: Math. Theor. 43 115303 which covers the cases of the
su(1,1) algebra (for x > 0), the su(2) algebra (for x < 0) and the h(4)
ordinary Weyl-Heisenberg algebra (for x = 0). For finite-dimensional
representations of A(x) and A(x,s), where A(x,s) is a truncation of order s of
A(x) in the sense of Pegg-Barnett, a connection is established with k-fermionic
algebras (or quon algebras). This connection makes it possible to use
generalized Grassmann variables for constructing certain coherent states.
Coherent states of the Perelomov type are derived for infinite-dimensional
representations of A(x) and for finite-dimensional representations of A(x) and
A(x,s) through a Fock-Bargmann analytical approach based on the use of complex
(or bosonic) variables. The same approach is applied for deriving coherent
states of the Barut-Girardello type in the case of infinite-dimensional
representations of A(x). In contrast, the construction of \`a la
Barut-Girardello coherent states for finite-dimensional representations of A(x)
and A(x,s) can be achieved solely at the price to replace complex variables by
generalized Grassmann (or k-fermionic) variables. Some of the results are
applied to su(2), su(1,1) and the harmonic oscillator (in a truncated or not
truncated form).Comment: 25 page
Quadratic Discrete Fourier Transform and Mutually Unbiased Bases
36 pages, submitted for publication in "Fourier Transforms, Theory and Applications", G. Nikolic (Ed.), InTech (Open Access Publisher), Vienna, 2011 - ISBN 978-953-307-231-9The present chapter [submitted for publication in "Fourier Transforms, Theory and Applications", G. Nikolic (Ed.), InTech (Open Access Publisher), Vienna, 2011] is concerned with the introduction and study of a quadratic discrete Fourier transform. This Fourier transform can be considered as a two-parameter extension, with a quadratic term, of the usual discrete Fourier transform. In the case where the two parameters are taken to be equal to zero, the quadratic discrete Fourier transform is nothing but the usual discrete Fourier transform. The quantum quadratic discrete Fourier transform plays an important role in the field of quantum information. In particular, such a transformation in prime dimension can be used for obtaining a complete set of mutually unbiased bases