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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 which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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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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