# Bosonic Representation of Matrices and Angular Momentum Probabilistic Representation of Cyclic States

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## Abstract

**:**

## 1. Introduction

## 2. Jordan–Schwinger Map of Lie Algebras

#### 2.1. Preliminaries

#### 2.2. Jordan–Schwinger Map of N-Dimensional Matrices

#### 2.3. Example: Bosonic Representation of $su$ (3) Algebra

## 3. Bosonic Representation of $\mathbf{su}$(2) Algebra and Applications

## 4. Probabilistic Representation of Cyclic States

## 5. Summary and Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Husimi-like $Q\left(\nu \right)$ probability distribution for the coherent state (black) and the cyclic states, even (gray) and odd (black, dashed), associated with the ${C}_{2}$ group. Here, the parameters $\alpha =1$, $\beta =i/5$, and $\nu =\rho {e}^{i\varphi}$.

**Figure 2.**Husimi-like $Q\left(\nu \right)$ probability distribution for the coherent state (black), cyclic states with irreducible representation $\lambda =1$ (gray), $\lambda =2$ (black, dashed), and $\lambda =3$ (gray, dashed), associated with the ${C}_{3}$ group. Here, the parameters $\alpha =1$, $\beta =i/5$, and $\nu =\rho {e}^{i\varphi}$.

**Figure 3.**Tomographic representation for the cyclic states associated with the cyclic group ${C}_{3}$ with irreducible representations $\lambda =1,2,3$ (

**a**–

**c**, respectively) associated with the ${C}_{3}$ group; here, the parameters $\alpha =2$ and $\beta =i$. Tomographic representation for $\lambda =1$, $\alpha =2$, $\beta =2i$ (

**d**). For all the cases, the parameters ${\mu}_{1}=cos(1/10)$, ${\nu}_{1}=sin(1/10)$, ${\mu}_{2}=cos(1/5)$, and ${\nu}_{2}=sin(1/5)$.

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**MDPI and ACS Style**

López-Saldívar, J.A.; Man’ko, O.V.; Man’ko, M.A.; Man’ko, V.I.
Bosonic Representation of Matrices and Angular Momentum Probabilistic Representation of Cyclic States. *Entropy* **2023**, *25*, 1628.
https://doi.org/10.3390/e25121628

**AMA Style**

López-Saldívar JA, Man’ko OV, Man’ko MA, Man’ko VI.
Bosonic Representation of Matrices and Angular Momentum Probabilistic Representation of Cyclic States. *Entropy*. 2023; 25(12):1628.
https://doi.org/10.3390/e25121628

**Chicago/Turabian Style**

López-Saldívar, Julio A., Olga V. Man’ko, Margarita A. Man’ko, and Vladimir I. Man’ko.
2023. "Bosonic Representation of Matrices and Angular Momentum Probabilistic Representation of Cyclic States" *Entropy* 25, no. 12: 1628.
https://doi.org/10.3390/e25121628