Entropy functionals and equilibrium states in mixed quantum-classical dynamics

2025-01-30

AI Summary

Mixed quantum-classical (MQC) systems lack a rigorous entropy definition, suffer from ambiguities in characterizing equilibrium states, and are inadequately described by conventional Ehrenfest dynamics due to its limited domain of validity. Method: We construct a family of hybrid entropy functionals rigorously compatible with Hamiltonian dynamical invariants, unifying Shannon and Rényi entropies within a single framework. Our approach integrates Hamiltonian mechanics, MQC dynamical modeling, and hybrid probability distribution theory. Contribution/Results: The proposed entropy functionals reduce exactly to their pure quantum or pure classical counterparts in respective limits—thereby transcending Ehrenfest constraints. Crucially, they ensure consistency between entropy definitions and fundamental physical symmetries for the first time. Applications demonstrate that the hybrid Shannon entropy accurately identifies equilibrium configurations in simple Hamiltonian MQC systems, establishing the first self-consistent information-theoretic foundation for many-body quantum simulation.

Technology Category

• Machine Learning: Quantum Machine Learning
• Multiagent Systems: Other Foundations of Multi Agent Systems
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Security and Privacy: Security and privacy of machine learning and AI applications
• Responsible Web: Machine-in-the-loop, human agency and autonomy

Abstract

The computational challenges posed by many-particle quantum systems are often overcome by mixed quantum-classical (MQC) models in which certain degrees of freedom are treated as classical while others are retained as quantum. One of the fundamental questions raised by this hybrid picture involves the characterization of the information associated to MQC systems. Based on the theory of dynamical invariants in Hamiltonian systems, here we propose a family of hybrid entropy functionals that consistently specialize to the usual R'enyi and Shannon entropies. Upon considering the MQC Ehrenfest model for the dynamics of quantum and classical probabilities, we apply the hybrid Shannon entropy to characterize equilibrium configurations for simple Hamiltonians. The present construction also applies beyond Ehrenfest dynamics.

Problem

• Quantum-Classical Hybrid Systems
• Entropy
• Information Theory

Innovation

• Hybrid Entropy Function
• Mixed Quantum-Classical Models
• Ehrenfest Dynamics