# On the Significance of the Quantum Mechanical Covariance Matrix

^{1}

^{2}

^{*}

## 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

- Navascués, M.; Guryanova, Y.; Hoban, M.J.; Acin, A. Almost quantum correlations. Nat. Commun.
**2015**, 6, 6288. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Popescu, S.; Rohrlich, D. Quantum nonlocality as an axiom. Found. Phys.
**1994**, 24, 379–385. [Google Scholar] [CrossRef] [Green Version] - Navascués, M.; Pironio, S.; Acín, A. A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations. New J. Phys.
**2008**, 10, 073013. [Google Scholar] [CrossRef] [Green Version] - Pawłowski, M.; Paterek, T.; Kaszlikowski, D.; Scarani, V.; Winter, A.; Żukowski, M. Information causality as a physical principle. Nature
**2009**, 461, 1101–1104. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Navascués, M.; Wunderlich, H. A glance beyond the quantum model. Proc. R. Soc. A
**2010**, 466, 881–890. [Google Scholar] [CrossRef] [Green Version] - Popescu, S. Nonlocality beyond quantum mechanics. Nat. Phys.
**2014**, 10, 264–270. [Google Scholar] [CrossRef] - Brunner, N.; Cavalcanti, D.; Pironio, S.; Scarani, V.; Wehner, S. Bell nonlocality. Rev. Mod. Phys.
**2014**, 86, 419. [Google Scholar] [CrossRef] - Goh, K.T.; Kaniewski, J.; Wolfe, E.; Vértesi, T.; Wu, X.; Cai, Y.; Liang, Y.-C.; Scarani, V. Geometry of the set of quantum correlations. Phys. Rev. A
**2018**, 97, 022104. [Google Scholar] [CrossRef] [Green Version] - Carmi, A.; Cohen, E. Relativistic independence bounds nonlocality. arXiv, 2018; arXiv:1806.03607. [Google Scholar]
- Landau, L. Empirical two-point correlation functions. Found. Phys.
**1988**, 18, 449–460. [Google Scholar] [CrossRef] - Bell, J.S. On the Einstein Podolsky Rosen paradox. Physics
**1964**, 1, 195–200. [Google Scholar] [CrossRef] [Green Version] - Clauser, J.F.; Horne, M.A.; Shimony, A.; Holt, R.A. Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett.
**1969**, 23, 880. [Google Scholar] [CrossRef] - Carmi, A.; Moskovich, D. Tsirelson’s bound prohibits communication through a disconnected channel. Entropy
**2018**, 20, 151. [Google Scholar] [CrossRef] - Tsirel’son, B.S. Quantum analogues of the Bell inequalities. The case of two spatially separated domains. J. Sov. Math.
**1987**, 36, 557–570. [Google Scholar] [CrossRef] - Masanes, L. Necessary and sufficient condition for quantum-generated correlations. arXiv, 2003; arXiv:quant-ph/0309137. [Google Scholar]
- Fine, A. Hidden variables, joint probability, and the Bell inequalities. Phys. Rev. Lett.
**1982**, 48, 291. [Google Scholar] [CrossRef] - Tsallis, C. Possible generalization of Boltzmann-Gibbs statistics. J. Stat. Phys.
**1988**, 52, 479–487. [Google Scholar] [CrossRef] - Oppenheim, J.; Wehner, S. The uncertainty principle determines the nonlocality of quantum mechanics. Science
**2010**, 330, 1072–1074. [Google Scholar] [CrossRef] [PubMed] - Aharonov, Y.; Albert, D.; Vaidman, L. How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100. Phys. Rev. Lett.
**1988**, 60, 1351–1354. [Google Scholar] [CrossRef] [PubMed] - Aharonov, Y.; Cohen, E.; Elitzur, A.C. Can a future choice affect a past measurement’s outcome? Ann. Phys.
**1988**, 355, 258–268. [Google Scholar] [CrossRef] - White, T.C.; Mutus, J.Y.; Dressel, J.; Kelly, J.; Barends, R.; Jeffrey, E.; Sank, D.; Megrant, A.; Campbell, B.; Chen, Y.; et al. Preserving entanglement during weak measurement demonstrated with a violation of the Bell-Leggett-Garg inequality. NPJ Quantum Inf.
**2016**, 2, 15022. [Google Scholar] [CrossRef] - Mermin, N.D. Extreme quantum entanglement in a superposition of macroscopically distinct states. Phys. Rev. Lett.
**1990**, 65, 1838. [Google Scholar] [CrossRef] [PubMed] - Collins, D.; Gisin, N. A relevant two qubit Bell inequality inequivalent to the CHSH inequality. J. Phys. A Math. Gen.
**2004**, 37, 1775–1787. [Google Scholar] [CrossRef] [Green Version] - Pál, F.K.; Vértesi, T. Maximal violation of a bipartite three-setting, two-outcome Bell inequality using infinite-dimensional quantum systems. Phys. Rev. A
**2010**, 82, 022116. [Google Scholar] [CrossRef]

**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.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

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.
https://doi.org/10.3390/e20070500

**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