# The Quantum-Classical Transition as an Information Flow

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## Abstract

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**PACS**89.70.Cf (Entropy and other measures of information); 05.45.Tp (Time series analysis); 03.65.Sq (Semiclassical theories and applications); 05.45.Mt (Quantum chaos; semiclassical methods)

## 1. Introduction

## 2. The CLQM for a Special Semi-Classical Model

## 3. Symbolic Transfer Entropy

## 4. Results

**Figure 1.**a) The directionality index ${T}^{S}$ vs. ${E}_{r}$ and b) ${T}_{q,c}^{S}$ and ${T}_{c,q}^{S}$ vs. ${E}_{r}$, for a wide ${E}_{r}-$range. We took $q\equiv \langle {x}^{2}\rangle $ and $c\equiv A$. The classical variable A is dominant across most of the range, except for small ${E}_{r}-$values, for which Uncertainty Principle becomes important enough that the quantal variable $\langle {x}^{2}\rangle $ becomes dominant. Note the absolute minimum of ${T}^{S}$ at ${{E}_{r}}^{cl}=21.55264$, beginning of the transition region.

**Figure 2.**a) The directionality index ${T}^{S}$ vs. ${E}_{r}$ and b) ${T}_{q,c}^{S}$ and ${T}_{c,q}^{S}$ vs. ${E}_{r}$, for an ${E}_{r}-$range that allows to visualize the three zones of the process, i.e., quantal, transitional, and classic, delimited, respectively, by ${{E}_{r}}^{\mathcal{P}}=3.3282$, and ${{E}_{r}}^{cl}=21.55264$. We took $q\equiv \langle {x}^{2}\rangle $ and $c\equiv A$ as in Figure 1. Note the absolute minimum of ${T}^{S}$ at ${{E}_{r}}^{cl}$, the local maximum at ${{E}_{r}}^{\mathcal{P}}$, and the absolute maximum close by (${E}_{r}\simeq 2.2$). Symmetric information flow obtains at ${E}_{r}^{M}=6.81$ (where the Statistical Complexity attains a maximum), well within the transition region. Classical variable A is the “leading” one from $+\infty $ until this point. For smaller ${E}_{r}-$values, $\langle {x}^{2}\rangle $ becomes dominant.

**Figure 3.**The directionality index ${T}^{S}$ vs. ${E}_{r}$ and ${T}_{q,c}^{S}$ and ${T}_{c,q}^{S}$ vs. ${E}_{r}$ (inset). Here we took $q\equiv \langle {x}^{2}\rangle $ and $c\equiv {P}_{A}$. The three stages of the process are visible between ${{E}_{r}}^{\mathcal{P}}=3.3282$ and ${{E}_{r}}^{cl}=21.55264$. Note the ${T}^{S}-$absolute maximum at ${{E}_{r}}^{\mathcal{P}}$. ${E}_{r}^{M}$ is here slightly off the mark (see text).

## 5. Conclusions

## Acknowledgements

## References and Notes

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- A flow of information takes place in both directions. However, the two flows do not have the same strength, i.e., more information is transferred from the classical to the quantal variables than vice versa. In this sense, we can assert that one variable is influencing the behavior of the other.

© 2010 by the authors; licensee Molecular Diversity Preservation International, 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/.

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**MDPI and ACS Style**

Kowalski, A.M.; Martin, M.T.; Zunino, L.; Plastino, A.; Casas, M.
The Quantum-Classical Transition as an Information Flow. *Entropy* **2010**, *12*, 148-160.
https://doi.org/10.3390/e12010148

**AMA Style**

Kowalski AM, Martin MT, Zunino L, Plastino A, Casas M.
The Quantum-Classical Transition as an Information Flow. *Entropy*. 2010; 12(1):148-160.
https://doi.org/10.3390/e12010148

**Chicago/Turabian Style**

Kowalski, Andres M., Maria T. Martin, Luciano Zunino, Angelo Plastino, and Montserrat Casas.
2010. "The Quantum-Classical Transition as an Information Flow" *Entropy* 12, no. 1: 148-160.
https://doi.org/10.3390/e12010148