# Entanglement Dynamics and Classical Complexity

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Analytical Results

## 3. Numerical Results

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Derivation of Equation (24)

## References

- Dowling, J.P.; Milburn, G.J. Quantum technology: The second quantum revolution. Phil. Trans. R. Soc. A
**2003**, 361, 1655–1674. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Wang, J.; Sciarrino, F.; Laing, A.; Thompson, M.G. Integrated photonic quantum technologies. Nat. Photonics
**2020**, 14, 273–284. [Google Scholar] [CrossRef] - Benenti, G.; Casati, G.; Rossini, D.; Strini, G. Principles of Quantum Computation and Information (A Comprehensive Textbook); World Scientific Singapore: Singapore, 2019. [Google Scholar]
- Cornfeld, I.P.; Fomin, S.V.; Sinai, Y.G. Ergodic Theory; Springer: New York, NY, USA, 1982. [Google Scholar] [CrossRef]
- Gu, Y. Evidences of classical and quantum chaos in the time evolution of nonequilibrium ensembles. Phys. Lett.
**1990**, 149, 95–100. [Google Scholar] [CrossRef] - Ford, J.; Mantica, G.; Ristow, G.H. The Arnol’d cat: Failure of the correspondence principle. Phys. Nonlinear Phenom.
**1991**, 50, 493–520. [Google Scholar] [CrossRef] - Gu, Y.; Wang, J. Time evolution of coarse-grained entropy in classical and quantum motions of strongly chaotic systems. Phys. Lett.
**1997**, 229, 208–216. [Google Scholar] [CrossRef] - Pattanayak, A.K.; Brumer, P. Chaos and Lyapunov exponents in classical and quantal distribution dynamics. Phys. Rev. E
**1997**, 56, 5174–5177. [Google Scholar] [CrossRef] [Green Version] - Sokolov, V.V.; Zhirov, O.V.; Benenti, G.; Casati, G. Complexity of quantum states and reversibility of quantum motion. Phys. Rev. E
**2008**, 78, 046212. [Google Scholar] [CrossRef] [Green Version] - Benenti, G.; Casati, G. How complex is quantum motion? Phys. Rev. E
**2009**, 79, 025201. [Google Scholar] [CrossRef] [Green Version] - Balachandran, V.; Benenti, G.; Casati, G.; Gong, J. Phase-space characterization of complexity in quantum many-body dynamics. Phys. Rev. E
**2010**, 82, 046216. [Google Scholar] [CrossRef] - Prosen, T. Complexity and nonseparability of classical Liouvillian dynamics. Phys. Rev. E
**2011**, 83, 031124. [Google Scholar] [CrossRef] [Green Version] - Benenti, G.; Carlo, G.G.; Prosen, T. Wigner separability entropy and complexity of quantum dynamics. Phys. Rev. E
**2012**, 85, 051129. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Qin, P.; Wang, W.G.; Benenti, G.; Casati, G. Complexity and instability of quantum motion near a quantum phase transition. Phys. Rev. E
**2014**, 89, 032120. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Rozenbaum, E.B.; Ganeshan, S.; Galitski, V. Lyapunov Exponent and Out-of-Time-Ordered Correlator’s Growth Rate in a Chaotic System. Phys. Rev. Lett.
**2017**, 118, 086801. [Google Scholar] [CrossRef] [Green Version] - Rammensee, J.; Urbina, J.D.; Richter, K. Many-Body Quantum Interference and the Saturation of Out-of-Time-Order Correlators. Phys. Rev. Lett.
**2018**, 121, 124101. [Google Scholar] [CrossRef] [PubMed] [Green Version] - García-Mata, I.; Saraceno, M.; Jalabert, R.A.; Roncaglia, A.J.; Wisniacki, D.A. Chaos Signatures in the Short and Long Time Behavior of the Out-of-Time Ordered Correlator. Phys. Rev. Lett.
**2018**, 121, 210601. [Google Scholar] [CrossRef] [Green Version] - Bergamasco, P.D.; Carlo, G.G.; Rivas, A.M.F. Out-of-time ordered correlators, complexity, and entropy in bipartite systems. Phys. Rev. Res.
**2019**, 1, 033044. [Google Scholar] [CrossRef] [Green Version] - Prakash, R.; Lakshminarayan, A. Scrambling in strongly chaotic weakly coupled bipartite systems: Universality beyond the Ehrenfest timescale. Phys. Rev. B
**2020**, 101, 121108. [Google Scholar] [CrossRef] [Green Version] - Wang, J.; Benenti, G.; Casati, G.; Wang, W.G. Complexity of quantum motion and quantum-classical correspondence: A phase-space approach. Phys. Rev. Res.
**2020**, 2, 043178. [Google Scholar] [CrossRef] - Bennett, C.H.; Bernstein, H.J.; Popescu, S.; Schumacher, B. Concentrating partial entanglement by local operations. Phys. Rev. A
**1996**, 53, 2046–2052. [Google Scholar] [CrossRef] - Lerose, A.; Pappalardi, S. Bridging entanglement dynamics and chaos in semiclassical systems. Phys. Rev. A
**2020**, 102, 032404. [Google Scholar] [CrossRef] - Bianchi, E.; Hackl, L.; Yokomizo, N. Linear growth of the entanglement entropy and the Kolmogorov-Sinai rate. J. High Energy Phys.
**2018**, 2018, 25. [Google Scholar] [CrossRef] - Miller, P.A.; Sarkar, S. Signatures of chaos in the entanglement of two coupled quantum kicked tops. Phys. Rev. E
**1999**, 60, 1542–1550. [Google Scholar] [CrossRef] [PubMed] - Fujisaki, H.; Miyadera, T.; Tanaka, A. Dynamical aspects of quantum entanglement for weakly coupled kicked tops. Phys. Rev. E
**2003**, 67, 066201. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Jacquod, P. Semiclassical Time Evolution of the Reduced Density Matrix and Dynamically Assisted Generation of Entanglement for Bipartite Quantum Systems. Phys. Rev. Lett.
**2004**, 92, 150403. [Google Scholar] [CrossRef] [Green Version] - Petitjean, C.; Jacquod, P. Lyapunov Generation of Entanglement and the Correspondence Principle. Phys. Rev. Lett.
**2006**, 97, 194103. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Husimi, K. Some formal properties of the density matrix. Proc. Phys. Math. Soc. Jpn.
**1940**, 22, 264. [Google Scholar] - Emerson, J.; Ballentine, L. Characteristics of quantum-classical correspondence for two interacting spins. Phys. Rev. A
**2001**, 63, 052103. [Google Scholar] [CrossRef] - Haake, F.; Gnutzmann, S.; Kuś, M. Quantum Signatures of Chaos; Springer: Heidelberg, Germany, 2018. [Google Scholar]

**Figure 1.**Three-dimensional Poincaré surface of section for the chaotic case (

**a**): $a=5,c=3$ and the near-integrable case (

**b**): $a=5,c=0.5$, where we fix ${\varphi}_{2}=0$. Here we only consider a single trajectory starting from $({\theta}_{1},{\varphi}_{1},{\theta}_{2},{\varphi}_{2})=(\frac{\pi}{4},0,\frac{3\pi}{4},0)$ (see text for the definition of the angles ${\theta}_{k}$ and ${\varphi}_{K}$, $k=1,2$).

**Figure 2.**Quantum (circles with solid line) and classical (triangles with solid line) averaged linear entropy for different ℏ and ${\hslash}_{c}$ in the kicked coupled tops model defined in Equation (29), for (

**a**): $a=5$, $c=3$ and (

**b**): $a=5$, $c=5$. The dashed lines indicate the functions $\overline{S}=({\lambda}_{1}+{\lambda}_{2})t$ (red) and $\overline{S}={\lambda}_{1}t$ (light blue). The initial values of $S(t=0)$ are subtracted.

**Figure 3.**Same as in Figure 2, but for weaker coupling strength $c=0.5$, for which motion is quasi-integrable. The lines $\overline{S}\left(t\right)\propto logt$ and $\overline{S}\left(t\right)\propto log{t}^{2}$ are also drawn.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wang, J.; Dietz, B.; Rosa, D.; Benenti, G.
Entanglement Dynamics and Classical Complexity. *Entropy* **2023**, *25*, 97.
https://doi.org/10.3390/e25010097

**AMA Style**

Wang J, Dietz B, Rosa D, Benenti G.
Entanglement Dynamics and Classical Complexity. *Entropy*. 2023; 25(1):97.
https://doi.org/10.3390/e25010097

**Chicago/Turabian Style**

Wang, Jiaozi, Barbara Dietz, Dario Rosa, and Giuliano Benenti.
2023. "Entanglement Dynamics and Classical Complexity" *Entropy* 25, no. 1: 97.
https://doi.org/10.3390/e25010097