Interaction Wave Functions and Interaction-Based Coherence and Entanglement in Complex Adaptive Systems

Quantum coherent entanglement was observed in 1991 through optical interference. It has been a vital resource in quantum information technology. Subsequent research has revealed a mutual relationship between coherence and entanglement, demonstrating a connection between the two intriguing phenomena. However, understanding many-body coherent entanglement is challenging, particularly given that the underlying mechanism of the Einstein-Podolsky-Rosen (EPR) paradox—proposed initially in 1935 — remains unresolved. Our study explores the dynamics of many-agent systems using financial theories in complex adaptive systems (CAS). We apply our findings to uncover mechanisms behind many-body distributions as well as interaction-based coherence and entanglement in quantum many-body systems, a prime example of CAS. From a holistic complexity perspective, we define density momentum, force, and energy, establishing a mathematical relationship among density energy, interaction energy, and linear potential. Utilizing the Hamilton-Jacobi equation Schrödinger initially employed, we derived a nonlocal many-agent wave equation and identified two sets of eigenvalue wave functions for CAS. We find a unified framework that connects the many-agent wave equation, applicable to quantum many-body systems, with Schrödinger's original wave equation. It includes many-body interaction wave functions. The interaction-coherent entanglement state vectors are mathematically complete and diagonal within the framework of Heisenberg matrix mechanics. The pure state vectors enable significant advancements in many-body computations, enhancing our understanding of non-Gaussian distributions in complex quantum entanglement. In conclusion, interaction-coherent frequencies between the repulsive density force and attractive restoring force states generate interaction-coherent entangled states. Although requiring energy, interaction-coherent entanglement represents high-quality resources, suggesting future applications in quantum many-body computation and quantum information technology. PACS: 89.75.-k (Complex Systems); 03.65.Ud (Entanglement and Quantum Nonlocality); 89.65.Gh (Economics, Econophysics, Financial Markets, Business and Management); 74.20.Mn (Nonconventional mechanisms)