  A quantum logic gate of particular interest to both electrical engineers and
game theorists is the quantum multiplexer. This shared interest is due to the
facts that an arbitrary quantum logic gate may be expressed, up to arbitrary
accuracy, via a circuit consisting entirely of variations of the quantum
multiplexer, and that certain one player games, the history dependent Parrondo
games, can be quantized as games via a particular variation of the quantum
multiplexer. However, to date all such quantizations have lacked a certain
fundamental game theoretic property.
  The main result in this dissertation is the development of quantizations of
history dependent quantum Parrondo games that satisfy this fundamental game
theoretic property. Our approach also yields fresh insight as to what should be
considered as the proper quantum analogue of a classical Markov process and
gives the first game theoretic measures of multiplexer behavior.
