  In the context of quantum information theory, "quantization" of various
mathematical and computational constructions is said to occur upon the
replacement, at various points in the construction, of the classical
randomization notion of probability distribution 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.
