1,395 research outputs found
SU(2) and SU(1,1) Approaches to Phase Operators and Temporally Stable Phase States: Applications to Mutually Unbiased Bases and Discrete Fourier Transforms
We propose a group-theoretical approach to the generalized oscillator algebra
Ak recently investigated in J. Phys. A: Math. Theor. 43 (2010) 115303. The case
k > or 0 corresponds to the noncompact group SU(1,1) (as for the harmonic
oscillator and the Poeschl-Teller systems) while the case k < 0 is described by
the compact group SU(2) (as for the Morse system). We construct the phase
operators and the corresponding temporally stable phase eigenstates for Ak in
this group-theoretical context. The SU(2) case is exploited for deriving
families of mutually unbiased bases used in quantum information. Along this
vein, we examine some characteristics of a quadratic discrete Fourier transform
in connection with generalized quadratic Gauss sums and generalized Hadamard
matrices
On the diagonalization of the discrete Fourier transform
The discrete Fourier transform (DFT) is an important operator which acts on
the Hilbert space of complex valued functions on the ring Z/NZ. In the case
where N=p is an odd prime number, we exhibit a canonical basis of eigenvectors
for the DFT. The transition matrix from the standard basis to the canonical
basis defines a novel transform which we call the discrete oscillator transform
(DOT for short). Finally, we describe a fast algorithm for computing the
discrete oscillator transform in certain cases.Comment: Accepted for publication in the journal "Applied and Computational
Harmonic Analysis": Appl. Comput. Harmon. Anal. (2009),
doi:10.1016/j.acha.2008.11.003. Key words: Discrete Fourier Transform, Weil
Representation, Canonical Eigenvectors, Oscillator Transform, Fast Oscillator
Transfor
Self-adjoint symmetry operators connected with the magnetic Heisenberg ring
We consider symmetry operators a from the group ring C[S_N] which act on the
Hilbert space H of the 1D spin-1/2 Heisenberg magnetic ring with N sites. We
investigate such symmetry operators a which are self-adjoint (in a sence
defined in the paper) and which yield consequently observables of the
Heisenberg model. We prove the following results: (i) One can construct a
self-adjoint idempotent symmetry operator from every irreducible character of
every subgroup of S_N. This leads to a big manifold of observables. In
particular every commutation symmetry yields such an idempotent. (ii) The set
of all generating idempotents of a minimal right ideal R of C[S_N] contains one
and only one idempotent which ist self-adjoint. (iii) Every self-adjoint
idempotent e can be decomposed into primitive idempotents e = f_1 + ... + f_k
which are also self-adjoint and pairwise orthogonal. We give a computer
algorithm for the calculation of such decompositions. Furthermore we present 3
additional algorithms which are helpful for the calculation of self-adjoint
operators by means of discrete Fourier transforms of S_N. In our investigations
we use computer calculations by means of our Mathematica packages PERMS and
HRing.Comment: 13 page
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