Quantum representations and their applications. Second Edition

The formal quantum representations demonstrate significant capabilities for solving classification problems in studying dynamical behavior of stochastic systems. Among their features are entanglement by measuring, quantitative estimating the fit for a closed qubit system and observations to identify qubit representation parameters, the specialized spectral metric for comparing patterns of system behaviors, with the multidimensional scaling, cluster analysis and discriminant analysis being applied to the matrices of mutual distances calculated using the presented spectral metric. Hidden evolution periodicities during the observation time period are determined by means of the quantum systems spectral analysis to clarify the system behavior structure. Applied quantum representations are purely theoretical constructs and do not require for their implementation special computational tools, providing quantum calculations in the usual sense. These representations can be used for representing the Markovian processes with the aid of qubit (i.e. 2-state quantum) representations of staying in the corresponding states. Presented also is a mathematical model of related activity designed to obtain comparative assessments of the activities of complex technical systems operators, representing their activity through a probability matrix of transitions between the performed elementary operations. The feasibility of this model is due to the high efficiency of quantum spectral analysis of transition probability matrices in combination with quantum filtering when solving classification problems. This study showed that with a sufficient volume of empirical data, even a one-time manifestation of some regularity indicates its presence and the regularity index value can measure the reliability of such a conclusion. The presented mathematical tools provide higher efficiency than other relevant approaches, obtaining diagnostic conclusions using empirical data of much smaller volume and extracting more helpful information. The given numerous illustrative examples of practical applications, mainly civil aviation ones, clarify benefits of the proposed techniques. This book is intended for both specialists in the field of applied mathematics and other experts involved in solving diagnostic and simulation problems. Compared to the previous edition, this book expands its coverage with two new sections. The first presents an implementation of activity simulation based on a small sample of empirical data. The second reports the results of computer experiments that clarify the key features of the quantum likelihood method for qubit evolution frequency identification.