# Bell Length as Mutual Information in Quantum Interference

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Non-Locality: A Survey

_{k}≤ 1 and $\sum _{k}}{p}_{k}=1$, thus their correlation derives from the probabilities p

_{k}. Werner’s results suggest that mixed separable entangled states may exist that do not violate Bell inequalities! Entanglement is therefore not a very strong condition in order to characterize non-locality. On the other hand, from the point of view of quantum field theory (QFT), a result of this type seems to be unsurprising [11].

_{j}are the eigenvalues of the real symmetric matrix $U={T}^{T}T$, ${T}_{ij}=Tr\left[\rho \left({\sigma}_{i}\otimes {\sigma}_{j}\right)\right]$ being the correlation matrix, whilst the degree of entanglement was measured by using the relative entropy of entanglement defined as

_{AB}, an interesting measuring system of the degree of entanglement is provided by Hall’s quantum correlation distance [13]:

## 3. Quantum Inference as Non Classical Distribution

## 4. Bell Inequalities and Entropic Correlation Length

^{C}. As a consequence, also in the entropic approach of Bell’s inequalities here suggested, entropic quantities associated with the Bell length (20) cannot distinguish between the perfect anti-correlation of A' and B' as it appears in (33), and the perfect correlation of A' and B' as in (34). In other words, for any marginal model with two-outcome measurements one has

^{max}achieves a value of 3 on the standard CHSH scenario, and therefore does not allows a quantum-mechanical realization since it is beyond Tsirelson’s bound of 2√2. This example shows that a convex combination of two non-violating marginal models may violate an entropic inequality like (31) and suggests the non-linear character of the entropic length Bell inequality (42) which is associated with the non-Euclidean geometry described by the Bell length.

## 5. Conclusions

## Appendix: The Positive-Definite Nature of the Bell Length

## Acknowledgements

## Conflicts of Interest

## References

- Bell, J.S. On the Einstein Podolsky Rosen paradox. Physics
**1964**, 1, 195–200. [Google Scholar] - Bell, J.S. Speakable. and Unspeakable in Quantum Mechanics; Cambridge University Press: Cambridge, UK, 1987. [Google Scholar]
- Chiatti, L. Wave Function Structure and Transactional Interpretation. In Waves and Particles in Light and Matter; van der Meerwe, A., Garuccio, A., Eds.; Springer: Berlin-Heidelberg, Germany, 1994; pp. 181–187. [Google Scholar]
- Chiatti, L. Path integral and transactional interpretation. Found. Phys.
**1995**, 25, 481–490. [Google Scholar] [CrossRef] - Kastner, R. The New Transactional Interpretation of Quantum Theory: The Reality of Possibility; Cambridge University Press: Cambridge, UK, 2013. [Google Scholar]
- Caves, C.M.; Fuchs, C.A.; Schack, R. Unknown quantum states: The quantum de Finetti representation. J. Math. Phys.
**2002**, 43, 4537–4559. [Google Scholar] [CrossRef] - Von Baeyer, H.C. “Quantum Weirdness? It’s all in your mind”. Sci. Am.
**2013**, 308, 46–51. [Google Scholar] [CrossRef] [PubMed] - Clauser, J.F.; Horne, M.A. Experimental consequences of objective local theories. Phys. Rev. D
**1974**, 10, 526–535. [Google Scholar] [CrossRef] - Clauser, J.F.; Horn, M.A.; Shimony, A.; Holt, R.A. Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett.
**1969**, 23, 880–884. [Google Scholar] [CrossRef] - Werner, R.F. Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. Phys. Rev. A
**1989**, 40, 4277–4281. [Google Scholar] [CrossRef] [PubMed] - Buscemi, F.; Compagno, G. Non-locality and causal evolution in QFT. J. Phys. B: At. Mol. Opt. Phys.
**2006**, 39, 695–709. [Google Scholar] [CrossRef] - Horst, B.; Bartkiewicz, K.; Miranowicz, A. “Two-qubit mixed states more entangled than pure states: Comparison of the relative entropy of entanglement for a given nonlocality”. Phys. Rev. A
**2013**, 87, 042108. [Google Scholar] [CrossRef] - Hall, M.W. Correlation distance and bounds for mutual information. Entropy
**2013**, 15, 3698–3713. [Google Scholar] [CrossRef] - Licata, I.; Fiscaletti, D. Quantum Potential. Physics, Geometry, Algebra; Springer: Berlin-Heidelberg, Germany, 2014. [Google Scholar]
- Licata, I.; Fiscaletti, D. Bohm trajectories and Feynman paths at light of quantum entropy. Acta. Phys. Pol. B
**2014**, 4, 885–904. [Google Scholar] [CrossRef] - Sbitnev, V.I. Bohmian split of the Schrödinger equation onto two equations describing evolution of real functions. Kvantovaya. Magiya.
**2008**, 5, 1101–1111. [Google Scholar] - Sbitnev, V.I. Bohmian trajectories and the path integral paradigm. Complexified lagrangian mechanics. Int. J. Bifurc. Chaos
**2009**, 19, 2335–2346. [Google Scholar] [CrossRef] - Fiscaletti, D. A geometrodynamic entropic approach to Bohm’s quantum potential and the link with Feynman’s path integrals formalism. Quantum Matter
**2013**, 2, 122–131. [Google Scholar] [CrossRef] - Novello, M.; Salim, J.M.; Falciano, F.T. On a geometrical description of Quantum Mechanics. Int. J. Geom. Methods Mod. Phys.
**2011**, 8, 87–98. [Google Scholar] [CrossRef] - Fiscaletti, D.; Licata, I. Weyl geometries, Fisher information and quantum entropy in quantum mechanics. Int. J. Theor. Phys.
**2012**, 51, 3587–3595. [Google Scholar] [CrossRef] - Fiscaletti, D. Toward a geometrodynamic entropic approach to quantum entanglement and the perspectives on quantum computing. EJTP
**2013**, 10, 109–132. [Google Scholar] - Chaves, R.; Fritz, T. An entropic approach to local realism and noncontextuality. Phys. Rev. A
**2012**, 85, 032113. [Google Scholar] [CrossRef] - Resconi, G.; Licata, I.; Fiscaletti, D. Unification of quantum and gravity by non classical information entropy space. Entropy
**2013**, 15, 3602–3619. [Google Scholar] [CrossRef] - Schiff, L.I. Quantum Mechanics; McGraw-Hill: New York, NY, USA, 1968. [Google Scholar]

© 2014 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 license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Licata, I.; Fiscaletti, D.
Bell Length as Mutual Information in Quantum Interference. *Axioms* **2014**, *3*, 153-165.
https://doi.org/10.3390/axioms3020153

**AMA Style**

Licata I, Fiscaletti D.
Bell Length as Mutual Information in Quantum Interference. *Axioms*. 2014; 3(2):153-165.
https://doi.org/10.3390/axioms3020153

**Chicago/Turabian Style**

Licata, Ignazio, and Davide Fiscaletti.
2014. "Bell Length as Mutual Information in Quantum Interference" *Axioms* 3, no. 2: 153-165.
https://doi.org/10.3390/axioms3020153