Hidden stochastic, quantum and dynamic information of Markov diffusion process and its evaluation by an entropy integral measure under the impulse controls actions, applied to information observer
Hidden information emerges under impulse interactions with Markov diffusion process modeling interactive random environment. Impulse yes no action cuts Markov correlations revealing Bit of hidden information connected correlated states. Information appears phenomenon of interaction cutting correlations carrying entropy. Each inter action models Kronicker impulse, delta impulse models interaction between the Kronicker impulses. Each impulse step down action cuts maximum of impulse minimal entropy and impulse step up action transits cutting minimal entropy to each step up action of merging delta function. Delta step down action kills delivering entropy producing equivalent minimax information. The merging action initiates quantum microprocess. Multiple cutting entropy is converting to information micro macroprocess. Cutting impulse entropy integrates entropy functional EF along trajectories of multidimensional diffusion process. Information which delivers ending states of each impulse integrates information path functional IPF along process trajectories. Hidden information evaluates Feller kernel whose minimal path transforms Markov transition probability to probability of Brownian diffusion. Each transitive transformation virtually observes origin of hidden information probabilities correlated states. IPF integrates observing Bits along minimal path assembling information Observer. Minimax imposes variation principle on EF and IPF whose extreme equations describe observing micro and macroprocess which describes irreversible thermodynamics. Hidden information curries free information frozen from correlated connections. Free information binds observing micro macro processes in information macrodynamics. Each dynamic three free information composes triplet structures. Three structural triplets assemble information network. Triple networks free information cooperate information Observer.
Hidden information emerges under impulse interactions with Markov diffusion process modeling interactive random environment. Impulse yes no action cuts Markov correlations revealing Bit of hidden information connected correlated states. Information appears phenomenon of interaction cutting correlations carrying entropy. Each inter action models Kronicker impulse, delta impulse models interaction between the Kronicker impulses. Each impulse step down action cuts maximum of impulse minimal entropy and impulse step up action transits cutting minimal entropy to each step up action of merging delta function. Delta step down action kills delivering entropy producing equivalent minimax information. The merging action initiates quantum microprocess. Multiple cutting entropy is converting to information micro macroprocess. Cutting impulse entropy integrates entropy functional EF along trajectories of multidimensional diffusion process. Information which delivers ending states of each impulse integrates information path functional IPF along process trajectories. Hidden information evaluates Feller kernel whose minimal path transforms Markov transition probability to probability of Brownian diffusion. Each transitive transformation virtually observes origin of hidden information probabilities correlated states. IPF integrates observing Bits along minimal path assembling information Observer. Minimax imposes variation principle on EF and IPF whose extreme equations describe observing micro and macroprocess which describes irreversible thermodynamics. Hidden information curries free information frozen from correlated connections. Free information binds observing micro macro processes in information macrodynamics. Each dynamic three free information composes triplet structures. Three structural triplets assemble information network. Triple networks free information cooperate information Observer.
Bayesian integral functional measure of entropy-uncertainty (EF) on trajectories of Markov multi-dimensional diffusion process is cutting off by interactive impulses (controls). Each cutoff minimax of EF superimposes and entangles conjugated fractions in microprocess, enclosing the captured entropy fractions as source of an information unit. The impulse step-up action launches the unit formation and step-down action finishes it and brings energy from the interactive jump. This finite jump transfers the entangled entropy from uncertain Yes-logic to the certain-information No-logic information unit whose measuring at end of the cut kills final entropy-uncertainty and limits unit. The Yes-No logic holds Bit Participator creating elementary information observer without physical pre-law. Cooperating two units in doublet and an opposite directional information unit in triplet forms minimal stable structure. Information path functional (IPF) integrates multiple hidden information contributions along the cutting process correlations in information units of cooperating doublets-triplets, bound by free information, and enfolds the sequence of enclosing triplet structures in the information network (IN) that sequentially decreases the entropy and maximizes information. The IN bound triplets release free information rising information forces enable attracting new information unit and ordering it. While IPF collects the information units, the IN performs logical computing using doublet-triplet code. The IN different levels unite logic of quantum micro- and macro- information processes, composing quantum and/or classical computation.
What is information originating in observation? Until now it has no scientifically conclusive definition. Information is memorized entropy cutting in random observations which processing interactions. Randomness of various interactive observations is source of entropy as uncertainty. Observation under random 1-0 impulses probabilities reveals hidden correlation which connects Bayesian probabilities increasing each posterior correlation. That sequentially reduces relational entropy conveying probabilistic casualty with temporal memory of correlations which interactive impulse innately cuts. Within hidden correlation emerges reversible time space microprocess with conjugated entangled entropy which probing impulse intentionally cuts and memorizes information as certainty. Sequential interactive cuts integrates cutting information in information macroprocess with irreversible time course. Memorized information binds reversible microprocess within impulse with irreversible information macroprocess. Observer probes collect cutting information data bits of observing frequencies impulses. Each impulse cuts maximum of impulse minimal information performing dual max-min principle of converting process entropy to information through uncertain gap. Multiple naturally encoding bits moving in macroprocess join triplet macrounits which logically organize information networks encoding macrounits in structures enclosing triplet code. Network time space distributed structure self renews and cooperates information decreasing its complexity. Integrating process entropy functional and bits information in information path integral embraces variation minimax law which determines processes regularities. Solving problem mathematically describes micro macro processes, network, and invariant conditions of observer network self replication.
Interacting random field of probabilities links Kolmogorov law 0-1 and Bayesian probabilities observing Markov diffusion process under Yes-No actions of random impulse. These objective probabilities measure virtual probing impulses processing the interactions in observable process. The impulse observation increases each posteriori correlation reducing conditional entropy measures from finite uncertainty up to certainty of real impulse. The reduced entropy conveys probabilistic causality with time course and temporal memory in collecting correlations, which interactive impulse innately cuts exposing hidden process entropy. The natural cut of this entropy reveals information hidden in the correlation connections. Inside the merging probing impulse emerges reversible microprocess with yes-no conjugated entangled entropy, curvature and logical complexity. Within the impulse time interval, entanglement starts before its space is formed and ends with beginning the space during reversible relative time interval being small part of impulse time interval. Merging impulse curves and rotates the impulse interactive actions in microprocess whose space interval measures this transitive movement.The cutting entropy sequentially converting to information memorizes the probes logic in Bit, participating in next probe conversions and encoding which memorizes information causality. The complexity and mass appears after the space emerges from the entanglement.The cognition assembles common units through the multiple attraction and resonances at forming network (IN) of triplet hierarchy, which accept only units that concentrates and recognizes each triplet IN node. The ended triplet of hierarchical INs measures level of the observer intelligence. The synthesized optimal process minimizes the observations time in Artificial designed information Observer with intellectual searching logic.
Up to now both information and its connection to reality have not scientifically conclusive definitions neither implicit origin. They emerge in observing multiple impulses interactive yes-no actions modeling information Bits. The observed information process connects reality, information, and creates Observer. Interaction is fundamental reality building structure of Universe from quantum physics up to brain neurons interactive actions. Merging action and reaction joins probabilistic prior and posterior actions on edge of the observed predictability and begins microprocess. Its time of entanglement starts space interval composing two qubits or Bit of reversible logic in the emerging information process. The impulse interacting action curves impulse geometry creating asymmetrical logic Bit as logical Maxwell demon. With approaching probability one, the attracting interaction captures energy memorizing asymmetrical logic in information certain Bit. Such Bit is naturally extracted at minimal quality energy equivalent ln2 working as Maxwell Demon. The memorized impulse Bit and its free information self-organizes multiple Bits in triplets sructuring macroprocess. Memorized information binds reversible microprocess with irreversible information macroprocess along multidimensional observing process. The macroprocess self-forming triplet units UP attract new UP through free Information. Multiple UP adjoin hierarchical network whose free information produces new UP at higher level node and encodes triplets in multi-levels hierarchical organization. The interactive information dynamics assemble geometrical and information structures of cognition and intelligence in double spiral rotating code. Information Path Functional integrates multiple interactive dynamics in finite bits which observe and measure reality. The time and space of reality exists only as units of information as Wheeler predicted.
We develop an information-theoretic reconstruction of quantum dynamics based on inference over action space. The fundamental object is a density of action states encoding the multiplicity of dynamical alternatives between configurations. Maximum-entropy inference introduces a finite resolution scale in action, implying that sufficiently close action contributions are operationally indistinguishable. We show that this indistinguishability, together with probability normalization and action additivity, selects complex amplitudes and unitary evolution as the minimal continuous representation compatible with action additivity, probability normalization, and inference under finite resolution. Quantum interference and unitarity therefore emerge as consequences of these assumptions rather than independent postulates. From the resulting propagator, the Lagrangian, Hilbert-space structure, and Schrödinger equation follow as derived consequences. In the infinitesimal-time limit, action differences universally fall below the resolution scale, making coherent summation the minimal consistent description at every step. The numerical value of the action scale is fixed empirically and identified with ℏ.
In this manuscript, we study discrete-time dynamics of systems that arise in physics and information theory, and the measure of disorder in these systems known as dy- namical entropy. The study of dynamics in classical systems is done from two distinct viewpoints: random walks and dynamical systems. Random walks are probabilistic in nature and are described by stochastic processes. On the other hand, dynami- cal systems are described algebraically and deterministic in nature. The measure of disorder from either viewpoint is known as dynamical entropy. Entropy is an essential notion in physics and information theory. Motivated by the study of disorder for the positions and velocities of gas molecules, the notion of entropy was first introduced mathematically by Boltzmann near the end of the 19th Century and gives rise to the second law of thermodynamics. Almost eighty years after Boltzmann, Shannon became the father of the new field of information theory when he produced his groundbreaking works where he used entropy as a measure of information transfer between two sources. In the last two years of the 1950’s, Kolmogorov and Sinai extended the notions of Boltzmann to a dynamical entropy. The Kolmogorov-Sinai dynamical entropy gives a measure for the disorder of a system of particles (e.g. gas molecules) averaged over time, quantifying the uncertainty in the dynamics of a system. The advent of quantum mechanics and its pervasiveness in nature has required the development of non-commutative generalizations of dynamics and dynamical entropy to the quantum regime. Many of each have been proposed. In particular, we recall the definitions of quantum random walks, dynamical systems and Markov chains. We motivate each generalization by relating to its classical counterpart. Quantum dynamical entropy (QDE) is a generalization of the Kolmogorov-Sinai dynamical entropy to quantum mechanics. There have been numerous definitions for QDE beginning with that of Connes, Narnhofer and Thirring in 1987. We focus on the semi- classical approach given by Słomczyński and Życzkowski in 1994 and the quantum Markov chain approach which started with Accardi, Ohya and Watanabe in 1997. Linearity of a dynamical entropy means that the dynamical entropy of the n-fold composition of a dynamical map with itself is equal to n times the dynamical entropy of the map for every positive integer n. We show that the quantum dynamical entropy introduced by Słomczyński and Życzkowski is nonlinear in the time interval between successive measurements of a quantum dynamical system. This is in contrast to Kolmogorov-Sinai dynamical entropy for classical dynamical systems, which is linear in time. We also compute the exact values of quantum dynamical entropy for the Hadamard walk with varying Lüders-von Neumann instruments and partitions. In 1948, Shannon proved the Source Coding Theorem which gives upper and lower bounds on the minimal expected codeword length in terms of the entropy of a ran- dom variable. This theorem can be leveraged to give the minimal expected average codeword length for a string of symbols in terms of entropy rate, which can be in- terpreted as the dynamical entropy of a stochastic process. In 1994, Schumacher defined indeterminate-length quantum codes and proved a Quantum Source Coding Theorem. We introduce the notion of stochastic ensembles of pure states and give a novel representation in terms of quantum Markov chains. Moreover, this representa- tion allows us to extend the Quantum Source Coding Theorem, giving the minimal expected average codeword length of an indeterminate-length quantum code in terms of quantum dynamical entropy. This manuscript includes joint work with Dr. George Androulakis (Mathematics, University of South Carolina).