Complex Adaptation and Entanglement in an Open Non-Local Many-Agent Wave Equation

ABSTRACT
Experiments reveal quantum behavioral deviation from Newton's laws' predictions, confirming the violation of Bell's inequality and the quantum non-locality. These results necessitate the definition of a non-local force, establishment of the non-local many-agent wave equation, and revelation of the underlying mechanisms of non-Gaussian distribution formation in complex quantum entanglement. However, finding the wave equation and establishing its unified paradigm with Schrödinger's wave equation are challenging since the open interacting agent systems are typically complex systems. Here we show cumulative observables in a time interval represent a density force in a one-dimensional sensitive variable coordinate from the complexity perspective and discover the equation. It justifies interaction-coherent entanglement with the zero-order Bessel distribution by conservation of interaction coherence and concludes that interactively coherent entangled agents show an "active matter-like" property over a sensitive variable. The theory predicts interactively coherent entanglement is a high-quality resource, surpassing the superposition entanglement squeezed by traditional preparation.
Keywords: complex adaptive systems, density force dynamics, interaction-coherent entanglement, "active matter-like" property, interaction conservation, non-Gaussian distribution
PACS: 89.75.-k (Complex Systems); 03.65.Ud (Entanglement and Quantum Non-locality); 89.65.Gh (Economics, Econophysics, Financial Markets, Business and Management)
Methods
The violation of Bell's inequality and the loop-free test of local realism via Hardy's violation confirm quantum non-locality. In addition, Newton's second law cannot predict quantum many-agent dynamic behaviors in complex quantum systems, in which entangled many-agents interact and are typically non-local. The results necessitate the definition of the non-local force that drives non-local dynamic behaviors in complex quantum many-agent systems. From the holistic perspective of complexity, we simplify a high-dimensional quantum many-agent system to a one-dimensional one, define the non-local momentum and the density force by cumulative observables in a time interval, and find a non-local many-agent wave equation. Observables such as distribution or density can determine invisible electron's dynamic behavior in atoms in Heisenberg's matrix mechanics, which is mathematically equivalent to Schrödinger's wave equation [43]. It is applicable for us to define the density momentum and the density force in physics.
Data
The theoretical study is based on two types of data. One comes from the physical experiments published in Science, Nature, and a monograph [44][45][55]. The other is collected from Chinese stock markets, as shown in the publications [6][62][63].
Non-local density momentum, density force, and linear potential
A non-local force or forces drive non-local dynamic behaviors in an open complex many-agent system. According to the relationship between a restoring force and potential in physics, a linear potential has a non-local restoring force. The force is always directed to the potential zero point or center. The magnitude of this force is independent of position, displacement, velocity, and acceleration in a one-dimensional sensitive variable coordinate, violating Newton's second law. Therefore, the linear potential function is suitable for describing non-local many-agent behaviors in "spooky actions at a distance". Its dimension is consistent with energy and interaction energy in complex many-agent systems.
Energy (or frequency), momentum, cumulative observables (or density), and sensitive variables are basic physical variables in quantum mechanics. We can describe complex many-agent behaviors by non-local probability waves. We define the non-local density momentum, density force, and energy in complex many-agent systems. The density momentum is cumulative observables in a time interval, the density force is the momentum in the time interval, and the energy is a product of the density force and the sensitive variable in the complex many-agent systems. The dimension of the energy is consistent with that of linear potential.
In physics, there is a precedent for using density or accumulative observables in a time interval to represent the magnitude of force in a complex system. For example, the higher the density of magnetic field lines is, the stronger the force in a magnetic field. On the contrary, the sparser the magnetic field lines are, the weaker the force in the magnetic field is at the location (see Fig. 2a). Similarly, the greater the probability of an intraday cumulative trading volume is at a price in complex financial markets, the greater the trading momentum and the force (see Fig. 2b) [63].
Quantum chemistry also backs up the definition of intangible non-local force by cumulative observables in a time interval since electron density uniquely determines the energy of a quantum-mechanical system according to density functional theory [64]. "In 1964, Walter Kohn (together with Pierre Hohenberg) proved a theorem that it is enough to know the ground state density to determine all observable quantities of a stationary quantum system. One year later, he derived (together with Lu Sham) a set of equations, which can be used to determine the ground state density" [65].


(a) (b)
Fig. 2 Density represents the magnitude of non-local momentum and the force in complex systems
Note: (a) The higher the density of magnetic field lines is, the stronger the magnetic force; (b) The larger the cumulative trading volume is, the greater the trading momentum and the force in an intraday cumulative trading volume distribution over a price range (The horizontal coordinate represents price, and the vertical coordinate is cumulative trading volume, probability or density) [63].

Assumptions and the mathematical expressions
Cumulative observables can represent non-local momentum and the density force in complex quantum many-agent systems. We write assumptions and mathematical expressions as follows,
•Assumption Ⅰ: The non-local momentum Q at a sensitive variable point q is a cumulative observable m at the point q in a time interval t, defined mathematically by equation (1);
•Assumption Ⅱ: The non-local density force Fd,tt is the momentum Q in the time interval t, written mathematically by equation (2);
•Assumption Ⅲ: Energy is a product of the momentum force Fm,tt and the sensitive variable q, expressed explicitly by equation (3).
Q≡∂S(q,t)/∂q=〖m/t=m〗_t, (1)
F_(d,tt)≡Q(q,t)/t=〖m_t/t=m/t^2 =m〗_tt, (2)
E(q,t)=F_(d,tt)*q=qm_tt=q m/t^2 =q m_t/t, (3)
where S(q,t) is action; q is a sensitive variable, which can be an electronic field, a magnetic field, or a light field to which quantum many-agents are sensitive; m is a cumulative observable at a point q; mt is a cumulative observable m at the point q in a time interval t or the non-local density momentum Q; mtt is the density momentum mt in the time interval or the density force Fd,tt; energy E(q,t) is product of the force Fd,tt and the sensitive variable q; t is a time interval, usually choosing one for convenience.
Conclusions
An intrinsic unified relationship exists between the non-local many-agent wave equation and the Schrödinger wave equation. The Schrödinger wave equation is found in an energy-conservative system in which Newton's second law governs the dynamic behaviors. Otherwise, the non-local many-agent wave equation is uncovered if the cumulative observable in a time interval represents the density momentum and the density force.
The non-local many-agent wave equation has two sets of analytical wavefunctions in the sensitive variable coordinate with different formation mechanisms. One is a set of wave functions with restoring force eigenvalues or energy eigenvalues and the mechanism of the function formation. The results are consistent with those in the Schrödinger wave equation. The other is a set of wave functions with interaction-coherent eigenvalues and the mechanism. It measures intangible interaction by interaction-coherent eigenvalues and provides new insights into complex many-agent systems.
An interaction-coherent entanglement state obeys the law of conservation of interaction coherence. Particles show "active matter-like" or "adaptive intelligence-like" properties in an interactively coherent entangled state. Our theory predicts that an interactively coherent entanglement state is a high-quality entangled resource since the eigenfunctions are independent and unitary in Hilbert space. It has higher fidelity, stronger decoherent resistance than the superposition entanglement state, and the ability to self-adaptation or self-repair. It can provide theoretical criteria and technical guidance for future industrial production of high-quality entangled resources.

Acknowledgments
Bing-Hong Wang and Jiuchang Wei thank the funding from the National Natural Science Foundation of China (Grant No: 71874172, 72293573), respectively.