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

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## Abstract

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**2010**, 43, 115303. The case $\kappa \ge 0$ corresponds to the noncompact group SU(1,1) (as for the harmonic oscillator and the Pöschl-Teller systems) while the case $\kappa <0$ 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 ${\mathcal{A}}_{\kappa}$ 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.

**PACS**03.65.Fd; 03.65.Ta; 03.65.Ud; 02.20.Qs

## 1. Introduction

## 2. Generalized Oscillator Algebra

#### 2.1. The Algebra

#### 2.2. The Oscillator Algebra as a Lie Algebra

#### 2.3. Rotated Shift Operators for Su(2) and Su(1,1)

#### 2.3.1. The Su(2) Case

#### 2.3.2. The Su(1,1) Case

## 3. Phase Operators

#### 3.1. The Su(2) Case

#### 3.2. The Su(1,1) Case

## 4. Truncated Generalized Oscillator Algebra

## 5. Mutually Unbiased Bases

#### 5.1. Quantization of the Phase Parameter

#### 5.2. Connecting the Phase Operator with a Quantization Scheme

#### 5.3. Introduction of Mutually Unbiased Bases

## 6. Discrete Fourier Transforms

#### 6.1. Quantum Quadratic Discrete Fourier Transform

#### 6.2. Quadratic Discrete Fourier Transform

#### 6.2.1. Factorization of the Quadratic DFT

#### 6.2.2. Hadamard Matrices

#### 6.2.3. Trace Relations

#### 6.2.4. Diagonalization

#### 6.2.5. Link with the Cyclic Group

#### 6.2.6. Decomposition

#### 6.2.7. Weyl Pairs

#### 6.2.8. Link with a Lie Algebra

## 7. Closing Remarks

## Acknowledgments

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## Appendix: Properties of the Ordinary DFT Matrix

**F**, one will be naturally interested finding d-point functions ${\Phi}_{n}^{(d)}(m)$ that are “good” discrete counterparts for the Hermite-Gauss functions ${\Psi}_{n}(x)$ in (127); in particular that they be analytic and periodic functions of m. Mehta [56] has proposed the following functions:

**F**, satisfying (135). Labelling these eigenvectors by their four Fourier eigenvalues ${\phi}_{k}$, and within each of these eigenspaces ${\mathbb{C}}^{{N}_{{\phi}_{k}}}$ by $j=0,1,\dots ,{N}_{{\phi}_{k}}-1$, we denote them by $\left\{{{\rm Y}}_{\phantom{\rule{-0.166667em}{0ex}}({\phi}_{k},j)}(m)\right\}$, periodic in m modulo d [46,58]; and we assume that they are complete in ${\mathbb{C}}^{d}$ and thus have a dual basis $\left\{{\widehat{{\rm Y}}}_{\phantom{\rule{-0.166667em}{0ex}}({\phi}_{k},j)}(m)\right\}$ periodic in m, such that

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**MDPI and ACS Style**

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.
https://doi.org/10.3390/sym2031461

**AMA Style**

Atakishiyev NM, Kibler MR, Wolf KB.
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(3):1461-1484.
https://doi.org/10.3390/sym2031461

**Chicago/Turabian Style**

Atakishiyev, Natig M., Maurice R. Kibler, and Kurt Bernardo Wolf.
2010. "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* 2, no. 3: 1461-1484.
https://doi.org/10.3390/sym2031461