How chaotic is a state of a quantum system?

Casati, Giulio

Boon Leong Lan

The nature of the stochastic process defined by Bohm's momentum is elucidated for stationary energy eigenstates and nonstationary states in classically nonchaotic and chaotic Hamiltonian systems. In addition, the nature of the stochastic process defined by Bohm's quantum force is elucidated for stationary energy eigenstates in nonchaotic and chaotic systems, and for nonstationary states in nonchaotic systems. From these results, the following can be concluded. For stationary energy eigenstates, the process defined by the momentum is generically a stationary, Dirac-delta process in both nonchaotic and chaotic systems; in contrast, the process defined by the quantum force is nongeneric. For nonstationary states, the processes defined by the momentum and quantum force are both nongeneric in nonchaotic systems. Furthermore, for nonstationary states, the process defined by the momentum is, with a high level of confidence, a stationary, stable, brown ${(f}^{\ensuremath{-}2}$ power spectrum) process in the chaotic kicked pendulum. It is conjectured that this is also true for other chaotic systems. The preceding conclusion and conjecture complement those in [Phys. Rev. A 63, 042105 (2001)] for the process defined by the quantum force for nonstationary states in chaotic systems.

Jincheng Zhang

This paper presents a novel approach to generating multi-dimensional quantum states leveraging the principles of multi-scale chaotic dynamics. The core idea revolves around constructing a complex chaotic system through carefully designed multi-scale chaotic mappings. This system effectively harnesses the inherent sensitivity and complexity of chaotic systems to facilitate the creation of quantum superposition and entanglement, ultimately leading to the efficient generation of quantum states with desired multi-qubit properties. The method offers a significant advancement over existing techniques that are often limited to single- or two-state generation. We demonstrate a framework for controlling the dimensionality and multi-qubit nature of the generated quantum states by meticulously adjusting the parameters of the underlying chaotic maps. The theoretical framework is presented, including key equations and mathematical formulations, providing a robust foundation for understanding and implementing this approach. The generated states are characterized using appropriate metrics, demonstrating the potential for applications in quantum information processing and simulation. ---

I. Puha

A. G. Lazerson, Alexey Alexeevich Boikov

Jincheng Zhang

This paper introduces a novel algorithm for optimizing chaotic systems based on the principles of quantum entanglement. The core idea is to represent the system's parameters as quantum states and utilize entanglement to dynamically adjust these states, thereby enabling precise control over the chaotic behavior. We explore the potential of this approach to overcome limitations of traditional optimization techniques and provide a new methodology for managing complex chaotic systems. The proposed algorithm demonstrates promising results in optimizing specific chaotic systems, showcasing the potential of quantum mechanics to enhance optimization processes.