\(\mathrm{SU}(2)\) and \(\mathrm{SU}(1,1)\) approaches to phase operators and temporally stable phase states: Applications to mutually unbiased bases and discrete Fourier transforms (Q527541)

Summary: We propose a group-theoretical approach to the generalized oscillator algebra \(\mathcal A_\kappa\) recently investigated in [\textit{M. Daoud} and \textit{M. R. Kibler}, J. Phys. A, Math. Theor. 43, No. 11, Article ID 115303, 18 p. (2010; Zbl 1186.81052)]. The case \(\kappa\geq 0\) corresponds to the noncompact group \(\mathrm{SU}(1,1)\) (as for the harmonic oscillator and the Pöschl-Teller systems) while the case \(\kappa<0\) is described by the compact group \(\mathrm{SU}(2)\) (as for the Morse system). We construct the phase operators and the corresponding temporally stable phase eigenstates for \(\mathcal A_\kappa\) in this group-theoretical context. The \(\mathrm{SU}(2)\) 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.