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)
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)
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)
Quantum entanglement was once a subject of intense philosophical debate and regarded as a puzzling theoretical anomaly. However, it has now been empirically validated and is widely recognized as a fundamental feature of quantum mechanics. This review traces its development in chronological order, beginning with the Einstein–Podolsky–Rosen (EPR) paradox, which raised profound questions about the completeness of quantum theory, and John Stewart Bell’s subsequent theorem, which established a measurable framework to verify entanglement’s non-local nature. The discussion then turns to contemporary experimental and applied domains where entanglement serves as a critical resource. Notable examples include quantum teleportation, entanglement-enhanced metrology, quantum radar, and emerging quantum energy devices such as quantum batteries. The transition from theoretical dispute to practical application underscores how entanglement now enables tangible technologies—ranging from ultra-secure communication networks to highly precise sensors. Furthermore, potential interdisciplinary applications in fields such as quantum biology are also explored, suggesting new pathways for innovation. This review aims to bridge foundational concepts with current technological advances, illustrating the continuing impact of quantum entanglement on both scientific understanding and real-world engineering.
Quantum many-body systems exhibit highly complex interactions that give rise to emergent collective phenomena beyond the behavior of individual particles. One of the most intriguing challenges in modern quantum physics is understanding how isolated quantum systems evolve toward thermal equilibrium despite following fundamentally reversible quantum dynamics. Quantum entanglement has emerged as a central mechanism governing information propagation, thermalization, and the development of statistical equilibrium in isolated many-body systems. Simultaneously, recent advances in quantum information science have enabled researchers to investigate the relationship between entanglement dynamics, information scrambling, and emergent quantum behavior using computational and analytical approaches. This study proposes a Quantum Information Analytics Framework for investigating thermalization processes in isolated many-body quantum systems through the integration of entanglement dynamics, quantum information measures, statistical mechanics, and computational modeling. The proposed framework evaluates system evolution using multiple performance indicators, including entanglement entropy, information propagation, thermal equilibrium, quantum coherence, and computational stability. The findings indicate that quantum entanglement plays a fundamental role in driving thermalization while preserving the underlying unitary evolution of isolated quantum systems. The proposed analytical framework contributes to a deeper understanding of emergent quantum dynamics and provides valuable insights for quantum computing, quantum simulation, condensed matter physics, and next-generation quantum information technologies.
This paper explores the potential of utilizing entangled multi-body states for enhanced quantum simulation. Traditional quantum simulation methods face limitations when simulating complex quantum systems, primarily due to their inability to effectively represent multi-particle correlations. We propose a novel approach leveraging the inherent properties of entangled states to achieve a more accurate and efficient simulation of these systems. Specifically, we examine the construction of quantum computers incorporating multi-body entangled states and their application to simulating quantum field theories, a domain where complex many-body interactions are prevalent. The core idea centers on exploiting the correlations encoded within entangled states to reduce the computational complexity associated with simulating high-dimensional quantum systems. We present a theoretical framework outlining the key challenges and potential strategies for realizing this approach, focusing on the design of appropriate quantum circuits and the optimization of entanglement protocols. The ultimate goal is to demonstrate a pathway towards simulating intricate quantum phenomena that are currently intractable using conventional simulation techniques. The simulation method is based on mapping the problem to a quantum computer. The quantum computer performs the simulation based on the state of the qubits. The final result is the answer to the original problem. The key mathematical tool is the quantum circuit. The quantum circuit is a series of quantum gates applied to qubits. The quantum gates are unitary transformations. The unitary transformations are represented by matrices. The matrix is a square matrix. The matrix is a linear transformation. The linear transformation is a transformation that preserves linear relationships. The eigenvalues of the matrix are the solutions to the equation det(A - λI) = 0, where A is the matrix and λ is the eigenvalue and I is the identity matrix. The eigenvectors of the matrix are the vectors that are transformed to themselves by the matrix. The eigenvectors are the solutions to the equation (A - λI)v = 0, where A is the matrix, λ is the eigenvalue and v is the eigenvector. The eigenvalues and eigenvectors are important concepts in linear algebra.
We have seen, in Section 2.5, how the superposition principle leads to the existence of entangled states of two or more quantum systems. Such states are characterized by the existence of correlations between the systems, the form of which cannot be satisfactorily accounted for by any classical theory. These have played a central role in the development of quantum theory since early in its development, starting with the famous paradox or dilemma of Einstein, Podolsky, and Rosen (EPR). No less disturbing than the EPR dilemma is the problem of Schrödinger’s cat, an example of the apparent absurdity of following entanglement into the macroscopic world. It was Schrödinger who gave us the name entanglement; he emphasized its fundamental significance when he wrote, ‘I would call this not one but the characteristic trait of quantum mechanics, the one that enforces the entire departure from classical thought’. The EPR dilemma represents a profound challenge to classical reasoning in that it seems to present a conflict between the ideas of the reality of physical properties and the locality imposed by the finite velocity of light. This challenge and the developments that followed have served to refine the concept of entanglement and will be described in the first section of this chapter. We start by recalling that a state of two quantum systems is entangled if its density operator cannot be written as a product of density operators for the two systems, or as a probability-weighted sum of such products. For pure states, the condition for entanglement can be stated more simply: a pure state of two quantum systems is not entangled only if the state vector can be written as a product of state vectors for the two systems. In the discipline of quantum information, entanglement is viewed as a resource to be exploited. We shall find, both here and in the subsequent chapters, that our subject owes much of its distinctive flavour to the utilization of entanglement.
Quantum mechanics led us to reconsider the scope of physics and its building principles, such as the notions of realism and locality. More recently, quantum theory has changed in an equally dramatic manner our understanding of information processing and computation. On one hand, the fundamental properties of quantum systems can be harnessed to transmit, store, and manipulate information in a more efficient and secure way than possible in the realm of classical physics. On the other hand, the development of systematic procedures to manipulate systems of a large number of particles in the quantum regime, crucial to the implementation of quantum-based information processing, has triggered new possibilities in the exploration of quantum many-body physics and related areas. In this thesis, we present new results relevant to two important problems in quantum information science: the development of a theory of entanglement, intrinsically quantum correlations, and the exploration of the use of controlled quantum systems to the computation and simulation of quantum many-body phenomena. In the first part we introduce a new approach to the study of entanglement