# Simplification of the Gram Matrix Eigenvalue Problem for Quadrature Amplitude Modulation Signals

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## Abstract

**:**

## 1. Introduction

## 2. Basic Theory

#### 2.1. Quantum Signals and Measurements

#### 2.2. Error Probability, Mutual Information, and Holevo Capacity

#### 2.3. Gram Matrix

#### 2.4. Square-Root Measurement

#### 2.5. Coherent-State Signals

#### 2.6. Symmetric Signals

**Definition**

**1**

**.**Let $(G;\circ )$ be a finite group with the operation ∘. A set $\left\{|{\psi}_{i}\rangle \right|i\in G\}$ of quantum state signals is called (narrow-sense) group covariant with respect to the group $(G;\circ )$ if the following is the case:

**Proposition**

**1**

**.**A set of quantum state signals $\left\{|{\psi}_{i}\rangle \right|i\in G\}$ is narrow-sense group covariant with respect to $(G;\circ )$ if and only if the following is the case.

## 3. Eigenvalues and Eigenvectors of $\mathit{M}=\mathbf{4}\mathit{m}$-ary QAM Signals and Their Gram Matrix

#### 3.1. $4m$-ary QAM Signals

**Definition**

**2**

**.**Let $\{{\beta}_{1},{\beta}_{2},\dots ,{\beta}_{m}\}$ be any m-ary set of complex amplitudes for which its arguments lie in the range $0<\phi <\frac{\pi}{2}$. That is, the complex amplitudes correspond to points in the first quadrant. Here, ${\beta}_{k}\ne 0\phantom{\rule{4pt}{0ex}}(k=1,2,\dots ,m)$ and ${\beta}_{k}\ne {\beta}_{{k}^{\prime}}\phantom{\rule{4pt}{0ex}}(k\ne {k}^{\prime})$ are assumed. For each ${\beta}_{k}$, let ${\alpha}_{k}^{\left(1\right)}={\beta}_{k}$, ${\alpha}_{k}^{\left(2\right)}=\mathbf{i}{\beta}_{k}$, ${\alpha}_{k}^{\left(3\right)}=-{\beta}_{k}$, and ${\alpha}_{k}^{\left(4\right)}=-\mathbf{i}{\beta}_{k}$, where $\mathbf{i}=\sqrt{-1}$. Then, we call the following set of coherent states “$4m$-ary QAM coherent-state signals” ($4m$-ary QAM signals for short):

#### 3.2. Gram Matrix of $4m$-ary QAM Signals

#### 3.3. Decomposition of Submatrices

#### 3.4. Decomposition of Gram Matrix

#### 3.5. Eigenvalues and Eigenvectors of Gram Matrix

#### 3.6. Relation of the Results in the Relevant Literature

## 4. Examples for the Case of $\mathit{m}=\mathbf{2}$

#### 4.1. Submatrices ${A}_{i}$

#### 4.2. Case of $|{\beta}_{1}|=|{\beta}_{2}|=\gamma $

#### 4.3. Case of $arg\left({\beta}_{1}\right)=arg\left({\beta}_{2}\right)=\nu $

## 5. Numerical Experiments

#### 5.1. Von Neumann Entropy

#### 5.2. Error Probability

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

SRM | Square-root measurement; |

PSK | Phase shift keying; |

ASK | Amplitude shift keying; |

QAM | Quadrature amplitude modulation; |

AMPM | Amplitude-modulated phase-modulated; |

POVM | Positive operator-valued measure. |

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**Figure 2.**von Neumann entropy of 16QAM signals with respect to ${\left|\alpha \right|}^{2}$. The blue line is drawn by using the results in Section 3, while the red dots are plotted by using direct calculation of eigenvalues for the Gram matrix.

**Figure 3.**Error probability of 16QAM signals with respect to ${\left|\alpha \right|}^{2}$. The blue line is drawn by using the results in Section 3, while the red dots are plotted by using direct calculation of the matrix square-root for the Gram matrix.

Eigenvalues | Eigenvectors |
---|---|

${a}_{j}^{\left(1\right)}$ | ${\mathit{a}}_{\mathit{j}}^{\left(\mathbf{1}\right)}\otimes {\mathbf{\lambda}}_{\mathbf{1}}$ |

${a}_{j}^{\left(2\right)}$ | ${\mathit{a}}_{\mathit{j}}^{\left(\mathbf{2}\right)}\otimes {\mathbf{\lambda}}_{\mathbf{2}}$ |

${a}_{j}^{\left(3\right)}$ | ${\mathit{a}}_{\mathit{j}}^{\left(\mathbf{3}\right)}\otimes {\mathbf{\lambda}}_{\mathbf{3}}$ |

${a}_{j}^{\left(4\right)}$ | ${\mathit{a}}_{\mathit{j}}^{\left(\mathbf{4}\right)}\otimes {\mathbf{\lambda}}_{\mathbf{4}}$ |

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**MDPI and ACS Style**

Miyazaki, R.; Wang, T.; Usuda, T.S.
Simplification of the Gram Matrix Eigenvalue Problem for Quadrature Amplitude Modulation Signals. *Entropy* **2022**, *24*, 544.
https://doi.org/10.3390/e24040544

**AMA Style**

Miyazaki R, Wang T, Usuda TS.
Simplification of the Gram Matrix Eigenvalue Problem for Quadrature Amplitude Modulation Signals. *Entropy*. 2022; 24(4):544.
https://doi.org/10.3390/e24040544

**Chicago/Turabian Style**

Miyazaki, Ryusuke, Tiancheng Wang, and Tsuyoshi Sasaki Usuda.
2022. "Simplification of the Gram Matrix Eigenvalue Problem for Quadrature Amplitude Modulation Signals" *Entropy* 24, no. 4: 544.
https://doi.org/10.3390/e24040544