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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
The Fourier U(2) Group and Separation of Discrete Variables
The linear canonical transformations of geometric optics on two-dimensional
screens form the group , whose maximal compact subgroup is the Fourier
group ; this includes isotropic and anisotropic Fourier transforms,
screen rotations and gyrations in the phase space of ray positions and optical
momenta. Deforming classical optics into a Hamiltonian system whose positions
and momenta range over a finite set of values, leads us to the finite
oscillator model, which is ruled by the Lie algebra . Two distinct
subalgebra chains are used to model arrays of points placed along
Cartesian or polar (radius and angle) coordinates, thus realizing one case of
separation in two discrete coordinates. The -vectors in this space are
digital (pixellated) images on either of these two grids, related by a unitary
transformation. Here we examine the unitary action of the analogue Fourier
group on such images, whose rotations are particularly visible
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