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Symmetry 2010, 2(3), 1461-1484; doi:10.3390/sym2031461
Article

SU(2) and SU(1,1) Approaches to Phase Operators and Temporally Stable Phase States: Applications to Mutually Unbiased Bases and Discrete Fourier Transforms

1, 2,3,4,*  and 5
1 Instituto de Matemáticas, Universidad Nacional Autónoma de México, Av. Universidad s/n, Cuernavaca, Morelos 62251, Mexico 2 Université de Lyon, 37 rue du repos, 69361 Lyon, France 3 Université Claude Bernard and CNRS/IN2P3, 43 Bd du 11 Novembre 1918, F-69622 Villeurbanne, France 4 Institut de Physique Nucléaire, 43 Bd du 11 Novembre 1918, F-69622 Villeurbanne, France 5 Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Av. Universidad s/n, Cuernavaca, Morelos 62251, Mexico
* Author to whom correspondence should be addressed.
Received: 9 June 2010 / Revised: 8 July 2010 / Accepted: 9 July 2010 / Published: 12 July 2010
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Abstract

We propose a group-theoretical approach to the generalized oscillator algebra Aκ recently investigated in J. Phys. A: Math. Theor. 2010, 43, 115303. The case κ ≥ 0 corresponds to the noncompact group SU(1,1) (as for the harmonic oscillator and the Pöschl-Teller systems) while the case κ < 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 Aκ 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.
Keywords: phase operators; phase states; mutually unbiased bases; discrete Fourier transform phase operators; phase states; mutually unbiased bases; discrete Fourier transform
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).
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Atakishiyev, N.M.; Kibler, M.R.; Wolf, K.B. SU(2) and SU(1,1) Approaches to Phase Operators and Temporally Stable Phase States: Applications to Mutually Unbiased Bases and Discrete Fourier Transforms. Symmetry 2010, 2, 1461-1484.

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