# On the Significance of the Quantum Mechanical Covariance Matrix

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## Abstract

**:**

## 1. Introduction

## 2. Covariance-Based Certificate of Nonlocality

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Proof.**

#### The Covariance in Quantum Mechanics

**Theorem**

**2.**

**Proof.**

## 3. Nonlocality and Tsallis Entropy

**Theorem**

**3.**

**Proof.**

## 4. Verification Using Weak Measurements

## 5. Relation to the NPA Hierarchy

## 6. Tripartite Covariance

## 7. Further Generalization of the Covariance Matrix

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Quantum-like bounds on any statistical theory in Equation (3). The paler is the region, the larger is the difference $\left|{\mathcal{M}}_{12}-{\mathcal{M}}_{34}\right|$. The quantum bound on the two-point correlators, where this difference vanishes, is shown in dark blue. Classical correlators make the bounded square. In this figure, ${\mathcal{B}}_{x}\stackrel{\mathrm{def}}{=}{c}_{1}+{c}_{2}+{(-1)}^{x}({c}_{3}-{c}_{4})$ is a symmetry of the Bell–CHSH parameter.

**Figure 2.**Tsallis entropy $S\left(a\right)$ quantifies the extent of nonlocality in the Bell–CHSH experiment.

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Carmi, A.; Cohen, E. On the Significance of the Quantum Mechanical Covariance Matrix. *Entropy* **2018**, *20*, 500.
https://doi.org/10.3390/e20070500

**AMA Style**

Carmi A, Cohen E. On the Significance of the Quantum Mechanical Covariance Matrix. *Entropy*. 2018; 20(7):500.
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**Chicago/Turabian Style**

Carmi, Avishy, and Eliahu Cohen. 2018. "On the Significance of the Quantum Mechanical Covariance Matrix" *Entropy* 20, no. 7: 500.
https://doi.org/10.3390/e20070500