Complex adaptive learning is intelligent. It is adaptive, learns in feedback loops, and generates hidden patterns as many individuals, elements or particles interact in complex adaptive systems (CAS). CAS highlights adaptation in life and lifeless complex systems cutting across all traditional natural and social sciences disciplines. However, discovering a universal law in CAS and understanding the underlying mechanism of distribution formation, such as a non-Gauss distribution in complex quantum entanglement, remains highly challenging. Quantifying the uncertainty of CAS by probability wave functions, the authors explore the inherent logical relationship between Schrödinger's wave equation in quantum mechanics and Shi's trading volume-price wave equation in finance. Subsequently, the authors propose a non-localized wave equation in quantum mechanics if cumulative observable in a time interval represents momentum or momentum force in Skinner-Shi (reinforcement-frequency-interaction) coordinates. It reveals that the invariance of interaction as a universal law exists in quantum mechanics and finance. The theory shows that quantum entanglement is an interactively coherent state instead of a consequence of the superposition of coherent states. As a resource, quantum entanglement is non-separable, steerable, and energy-consumed. The entanglement state has opposite states subject to interaction conservation between the momentum and reversal forces. Keywords: complex adaptive systems, complex adaptive learning, complex quantum systems, non-localized wave equation, interaction conservation, interactively coherent entanglement PACS: 89.75.-k (Complex Systems); 89.65.Gh (Economics, Econophysics, Financial Markets, Business and Management); 03.65.Ud (Entanglement and Quantum Nonlocality)
Complex adaptive learning is intelligent. It is adaptive, learns in feedback loops, and generates hidden patterns as many individuals, elements or particles interact in complex adaptive systems (CAS). CAS highlights adaptation in life and lifeless complex systems cutting across all traditional natural and social sciences disciplines. However, discovering a universal law in CAS and understanding the underlying mechanism of distribution formation, such as a non-Gauss distribution in complex quantum entanglement, remains highly challenging. Quantifying the uncertainty of CAS by probability wave functions, the authors explore the inherent logical relationship between Schrödinger's wave equation in quantum mechanics and Shi's trading volume-price wave equation in finance. Subsequently, the authors propose a non-localized wave equation in quantum mechanics if cumulative observable in a time interval represents momentum or momentum force in Skinner-Shi (reinforcement-frequency-interaction) coordinates. It reveals that the invariance of interaction as a universal law exists in quantum mechanics and finance. The theory shows that quantum entanglement is an interactively coherent state instead of a consequence of the superposition of coherent states. As a resource, quantum entanglement is non-separable, steerable, and energy-consumed. The entanglement state has opposite states subject to interaction conservation between the momentum and reversal forces. Keywords: complex adaptive systems, complex adaptive learning, complex quantum systems, non-localized wave equation, interaction conservation, interactively coherent entanglement PACS: 89.75.-k (Complex Systems); 89.65.Gh (Economics, Econophysics, Financial Markets, Business and Management); 03.65.Ud (Entanglement and Quantum Nonlocality)
Abstract 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)
Adaptive wave model for financial option pricing is proposed, as a high-complexity alternative to the standard Black--Scholes model. The new option-pricing model, representing a controlled Brownian motion, includes two wave-type approaches: nonlinear and quantum, both based on (adaptive form of) the Schr\odinger equation. The nonlinear approach comes in two flavors: (i) for the case of constant volatility, it is defined by a single adaptive nonlinear Schr\odinger (NLS) equation, while for the case of stochastic volatility, it is defined by an adaptive Manakov system of two coupled NLS equations. The linear quantum approach is defined in terms of de Broglie's plane waves and free-particle Schr\odinger equation. In this approach, financial variables have quantum-mechanical interpretation and satisfy the Heisenberg-type uncertainty relations. Both models are capable of successful fitting of the Black--Scholes data, as well as defining Greeks.
Keywords: Black--Scholes option pricing, adaptive nonlinear Schr\odinger equation, adaptive Manakov system, quantum-mechanical option pricing, market-heat potential
PACS: 89.65.Gh, 05.45.Yv, 03.65.Ge
This paper explores the intersection of Complex Adaptive Systems (CAS) and whole-systems quantum computing, focusing on how CAS principles can enhance quantum computing methods and vice versa. CAS, characterized by complexity, adaptability, self-organization, and emergence, amongst other properties, are prevalent in various domains, including ecology, economics, and social systems. Whole-systems quantum computing, particularly through quantum annealing, manipulates the global quantum state of a system, utilizing entanglement and superposition to solve optimization problems. The paper outlines a methodology for modeling complex adaptive problems using quantum annealing, alongside case studies illustrating CAS theory’s application in quantum computation. By examining representative quantum computing companies the paper investigates their management of entropy-a key summary metric in CAS-and correlations with quantum supremacy. The integration of CAS dynamics into quantum computing is posited to aid in designing more advanced Hamiltonians and quantum algorithms, enhancing the capability to tackle complex, real-world problems more efficiently. The paper calls for further research into several promising avenues, such as the development of sophisticated Hamiltonians that encode complex interdependencies, the validation of CAS models on quantum hardware, and the potential for systematizing quantum supremacy through a virtuous cycle of CAS-informed quantum algorithm development.
Financial markets are complex nonlinear systems characterized by high-dimensional interactions, temporal dependencies, and rapidly evolving asymmetric patterns. Detecting hidden structures and abnormal behaviors in high-frequency financial data remains a challenging problem due to market instability and dynamic uncertainty. This study proposes an innovative quantum neural network (QNN)-based framework for adaptive forecasting and anomaly detection by exploiting quantum representation learning for complex financial systems. The proposed architecture is developed in the Qiskit environment using a hybrid classical-quantum optimization strategy based on a Variational Quantum Circuit (VQC). A quantum feature-mapping mechanism transforms multidimensional financial variables into quantum states, enabling extraction of latent relationships and structural patterns from high-dimensional data. A parameterized quantum circuit is optimized using the ADAM algorithm within a classical-quantum learning loop. The framework consists of two stages: first, multivariate financial dynamics are forecasted using market microstructure, volatility-related, and macroeconomic indicators obtained from the Electronic Data Distribution System (EDDS) of the Central Bank of the Republic of Türkiye; second, deviations between predicted and observed states are analyzed for anomaly identification. Comprehensive experiments on the BIST All-100 Index, trading volume, and monetary indicators demonstrate that the proposed QNN model achieves superior performance compared with ANN, LSTM, Bi-LSTM, LSTM-GRU, CNN, and Transformer-based approaches. The findings indicate that quantum-enhanced learning provides an effective computational framework for modeling complex asymmetric patterns and discovering hidden structures in dynamic systems.
This paper introduces a novel quantum model, the Adaptive Quantum Entanglement Model (AQEM), designed to simulate and predict complex quantum systems, particularly those exhibiting self-organization and self-adaptation. Traditional quantum models often lack the capacity to capture these dynamic, self-adjusting behaviors. The AQEM employs a dynamic adjustment of quantum entanglement strength and interaction to emulate self-adaptive systems, enabling more accurate predictions of system behavior. This research addresses a critical limitation in existing quantum simulation techniques by providing a mechanism for systems to adapt to changing conditions, offering a significant advancement in the modeling of complex quantum phenomena.
Abstract This paper extends the Local Tension Deficit (LTD) framework into Quantum Mechanics, postulating that quantum phenomena arise from the continuous, fluctuating nature of the Vacuum Energy Density (DEV). Quantum Uncertainty is interpreted as the physical noise imposed by DEV fluctuations on the particle's LTD structure. Wave Function Collapse is defined as the forced, instantaneous stabilization of the particle's LTD into a single, minimum potential energy configuration upon interaction. Quantum Entanglement is explained not as "action at a distance," but as the simultaneous compensatory adjustment of two or more LTD structures that share a single, non-local tension configuration within the DEV medium. 1. Introduction: The LTD Framework and the Quantum Scale 1.1. DEV and the Microcosmic Medium Previous works have established the Vacuum Energy Density (DEV) as the fundamental medium and the Local Tension Deficit (LTD)—created by matter's kinetic energy consuming DEV—as the source of all fundamental forces. At the quantum scale, the particle itself is viewed as a highly localized, persistent LTD structure. 1.2. Probability vs. Field Dynamics The probabilistic nature of Quantum Mechanics is here reinterpreted as the dynamic and noisy interaction between the particle's LTD structure and the highly fluctuating, underlying DEV field. 2. ❓ The Principle of Uncertainty: Noise from Vacuum Fluctuation The Uncertainty Principle dictates that pairs of conjugate variables (like position and momentum) cannot be known simultaneously with perfect precision. 2.1. Uncertainty as Physical Noise Quantum Uncertainty is not merely an observational limit; it is a physical reality imposed by the intrinsic tension fluctuations (noise) within the DEV field. The DEV Fluctuation Barrier: The high-frequency, continuous changes in vacuum tension make it impossible for the LTD structure of a particle to sustain a perfectly defined position (spatial stability) and a perfectly defined momentum (kinetic stability) simultaneously. The Position-Momentum Tradeoff: Any attempt to stabilize the LTD structure's position (by dampening the DEV fluctuations locally) inevitably leads to an increase in momentum fluctuation, and vice-versa, as the DEV field resists rigid constraints. 3. 💥 Wave Function Collapse: Forced LTD Stabilization The wave function (\Psi) describes the potential states of a particle before measurement. Collapse is the instantaneous transition to a single, definite state upon observation. 3.1. The Wave Function as Distributed Tension The Wave Function describes the spatial distribution of the particle's potential LTD configurations. Before measurement, the LTD structure of the particle is distributed across the DEV medium, corresponding to a superposition of potential energy states. 3.2. Measurement and Stabilization Measurement is defined as a strong, localized interaction between the particle's LTD structure and the complex LTD structures of the macroscopic measuring apparatus. Forced Transition: This interaction forces the distributed LTD of the particle to resolve into a single, definite structure that corresponds to the minimum potential energy state achievable within the local constraints imposed by the measurement. Collapse: The Wave Function Collapse is the instantaneous, non-reversible stabilization of the particle's LTD structure into a singular point in configuration space. 4. ♾️ Quantum Entanglement: The Shared Tension Structure Entanglement describes two or more particles linked such that measuring one instantly dictates the state of the others, regardless of distance. 4.1. Entanglement as Non-Local LTD Entangled particles are not two separate entities communicating faster than light; they are two or more highly localized LTD structures that belong to a single, shared, non-local tension configuration within the DEV medium. Single System: The vacuum medium treats the entangled pair as a single, combined structure of tension, which must always conserve its total angular momentum and potential energy. 4.2. Simultaneous Compensation When one particle's LTD structure is forced to stabilize (collapse) during a local measurement: The change instantly creates a massive tension gradient imbalance in the shared LTD structure. The DEV medium compensates simultaneously by forcing the distant LTD structure to instantaneously adopt the corresponding compensating state, maintaining the minimum potential energy for the entire, non-local system. This instantaneous correlation is a field adjustment, not a superluminal signal. 5. Conclusion: Unifying Quantum Mechanics with Field Dynamics This work establishes that the perplexing phenomena of Quantum Mechanics—Uncertainty, Collapse, and Entanglement—are direct, physical consequences of the continuous and dynamic interaction between the particle's Local Tension Deficit (LTD) structure and the highly energetic, fluctuating medium of the Vacuum Energy Density (DEV). This provides a coherent, non-probabilistic foundation for the quantum world, consistent with the unification of all other fundamental forces.