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

We study emergence of quantum and dynamic information during an observer’s impulse interactions with environment. The interactive delta-impulse of Markov diffusion process cuts the process correlation revealing hidden information connecting the process inner states. Information appears as phenomenon of interactions. The impulse probing probabilities, observing random process under Kolmogorov’s 1-0 law, link Bayesian [(0-1) or (1-0)] probabilities that increases each posterior correlation and reduces conditional entropy measures up to cutting this entropy by real impulse. Cutting random process’ correlation, covering interactive internal events, reveals underneath of each cutoff both hidden quantum and classical information. The cutting entropy sequentially converts to information and memorizes the impulse probabilistic logic in bit, which participates in next conversions and encoding. The impulse step-down action cuts a maximum of impulse’ minimal uncertainty-entropy, and impulse’ step-up action transforms the cutting portion to ending impulse state, where it kills the uncertainty, delivering equal minimax information. The interactive impulse holds maxminminimax law for transforming each minimum of a priori probability to maximum of a posteriori probability in the Bayesian link, and extracting information hidden in the cutting correlation in real cutoff. The applied impulse control cuts Markov multi-dimensional diffusion process, transforms each cutoff to Brownian diffusion and back to the Markovian, concurrently producing Feller’s kernel and initiates a Schrödinger’s bridge and an entanglement. An interactive jump at locality of Schrödinger’s bridge ‘edge in a sub-markov process creates microdynamic process within each impulse, which transforms the impulse time interval to space intervals preserving the impulse information measure. Entropy integral functional measures hidden information covering the process correlations for both the kernel and bridge along with integral time interval at the multiple cutoff transformations. The interactive processes imply transformation of each cutoff stochastic entropy portion to information bit of an emerging elementary information observer. The observer implements the maxmin extraction and minimax consumption as optimal information law for observing, selecting, transforming, and cooperating the cutting portions in multiple bits' units of information dynamics with a feedback to a receptive observer “window” where the interaction takes place. The law initiates minimax variation principle for the entropy functional whose variation equations determine the structure of the information dynamics arising at this transformation. The information path functional integrates multiple hidden information contributions of the cutting process correlations in information units, binds their information in doublets-triplets structures, and enfolds this sequence in the information network (IN) that successively decreases the structures’ entropy and maximizes information of the micro-macrodynamics. The IN enclosing triplets, through their “free” unbound information, rise information forces, attract, order, structure information units hierarchy, encode the memorized doublet-triplet logic, that composes quantum and classical computation. The IN ending triplets’ free information sequentially requests from the window and attaches the needed interactive information, and integrate the IN encoding information. The interactive observation unites uncertainty of random process and observer information process on the path to certainly and intelligence, while the paper math tools allows artificial design and compute this path. Keyword: hidden information; integral information measure; interactive impulse cutoff, impulse control; mnimax law, Feller kernel; Schrödinger bridge; probability symmetry, entropy functional, information path functional, cooperative micro-macro information dynamics, hierarchical network, logical computation. Introduction Uncovering and evaluation of quantum and dynamic information hidden in correlation of stochastic process through the impulse control presents new and essential topic in understanding the process of interaction an observer with its environment. Conventional information science generally considers an information process, but traditionally uses the probability measure for the random events and Shannon’s entropy measure as uncertainty function of the states [1, 2, 3]. Revealing information, covering a process’ inner connections between its states, within the hidden statistical and quantum dependencies along the process, is important problem in information theory [4, 5].