Design of Optimal Dynamic Analyzers: Mathematical Aspects of Wave Pattern Recognition

We give a review of the most important results on optimal tomography as mathematical wave-pattern recognition theory emerged in the 70's in connection with the problems of optimal estimation and hypothesis testing in quantum theory. In quantum theory such problems are sometimes referred as the problem of optimal measurement of an unknown quantum state, and are the main subject of the emerging mathematical theory of quantum statistics. We develop the results of quantum pattern recognition theory, most of which belong to VPB, further into the direction of wave, rather than particle statistical estimation and hypothesis testing theory, with the aim to include not only quantum matter waves but also classical wave patterns like optical and acoustic waves. We conclude that Hilbert space and operator methods developed in quantum theory are equally useful in the classical wave theory, as soon as the possible observations are restricted to only intensity distributions of waves, i.e. when the wave states are not the allowed observables, as they are not the observables of individual particles in the quantum theory. We show that all characteristic attributes of quantum theory such as complementarity, entanglement or Heisenberg uncertainty relations are also attributes of the generalized wave pattern recognition theory.