# Rényi Entropy, Signed Probabilities, and the Qubit

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

## 3. Rényi Entropy

**Definition**

**3.**

## 4. Main Theorem

This says that we allow as potential quantum states only those states $\mathit{r}$ containing a minimum amount of uncertainty, as measured by the entropy of a corresponding probability distribution $\mathit{q}$ on phase space. Note that our Uncertainty Principle is a sequence of conditions, one for each k. This is because Rényi entropy itself is not a single functional but a sequence of functionals (indexed by k).Uncertainty Principle: A potential quantum state $\mathit{r}$ satisfies the Uncertainty Principle if for every k, there is a phase-space probability distribution $\mathit{q}$ that represents $\mathit{r}$ and satisfies ${H}_{2k}\left(\mathit{q}\right)\ge 2$.

**Theorem**

**1.**

**Proof.**

**Claim**

**1.**

**Proof.**

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Claim**

**A1.**

- 1.
- If$\mathit{w}$is a solution then so is$\lambda \mathit{w}$for any$\lambda \ne 0$.
- 2.
- If$\mathit{w}$is a solution then$\mathit{v}$is a solution, where$\mathit{v}$is obtained from$\mathit{w}$by permuting coordinates.

**Proof.**

**Claim**

**A2.**

**Proof.**

**Claim**

**A3.**

**Proof.**

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**MDPI and ACS Style**

Brandenburger, A.; La Mura, P.; Zoble, S.
Rényi Entropy, Signed Probabilities, and the Qubit. *Entropy* **2022**, *24*, 1412.
https://doi.org/10.3390/e24101412

**AMA Style**

Brandenburger A, La Mura P, Zoble S.
Rényi Entropy, Signed Probabilities, and the Qubit. *Entropy*. 2022; 24(10):1412.
https://doi.org/10.3390/e24101412

**Chicago/Turabian Style**

Brandenburger, Adam, Pierfrancesco La Mura, and Stuart Zoble.
2022. "Rényi Entropy, Signed Probabilities, and the Qubit" *Entropy* 24, no. 10: 1412.
https://doi.org/10.3390/e24101412