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

TL;DR

This paper develops a group-theoretical framework for phase operators and stable phase states in generalized oscillator algebras, applying SU(2) and SU(1,1) groups to quantum information and Fourier transforms.

Contribution

It introduces a unified group-theoretical method to construct phase operators and states for generalized oscillators, linking them to mutually unbiased bases and Fourier transforms.

Findings

Constructed phase operators and stable states for Ak algebra.

Derived mutually unbiased bases using SU(2) approach.

Explored quadratic Fourier transforms and their mathematical properties.

Abstract

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.