Properly Quantized History Dependent Parrondo Games, Markov Processes, and Multiplexing Circuits

In the context of quantum information theory, “quantization ” of various mathemat-ical and computational constructions is said to occur upon the replacement, at various points in the construction, of the classical randomization notion of probability dis-tribution with higher order randomization notions from quantum mechanics such as quantum superposition with measurement. For this to be done “properly”, a faithful copy of the original construction is required to exist within the new “quantum ” one, just as is required when a function is extended to a larger domain. Here procedures for extending history dependent Parrondo games, Markov processes and multiplexing circuits to their ”quantum ” versions are analyzed from a game theoretic viewpoint, and from this viewpoint, proper quantizations developed.