Modeling quantum measurement probability as a classical stochastic process

The time-dependent measurement probabilities for the simple two-state quantum oscillator seem to invite description as a classical two-state stochastic process. It has been shown that such a description cannot be achieved using a Markov process. Constructing a more general non-Markov process is a challenging task, requiring as it does the proper generalizations of the Markovian Chapman-Kolmogorov and master equations. Here we describe those non-Markovian generalizations in some detail, and we then apply them to the two-state quantum oscillator. We devise two non-Markovian processes that correctly model the measurement statistics of the oscillator, we clarify a third modeling process that was proposed earlier by others, and we exhibit numerical simulations of all three processes. Our results illuminate some interesting though widely unappreciated points in the theory of non-Markovian stochastic processes. But since quantum theory does not tell us which one of these quite different modeling processes "really" describes the behavior of the oscillator, and also since none of these processes says anything about the dynamics of other (noncommuting) oscillator observables, we can see no justification for regarding any of these processes as being fundamentally descriptive of quantum dynamics. (C) 2001 American Institute of Physics.