Comment on "Quantum Measurements and Stochastic Processes"

Alberto Barchielli

Wilhelm von Waldenfels

L. de la Peña-Auerbach, R. M. Velasco, Ana Marı́a Cetto

Daniel T. Gillespie

It is argued that the stochastic model of a quantum-mechanical two-state oscillator discussed in the preceding Comment by Hardy et al. [Phys. Rev. A 56, 3301 (1997)] does not constitute a valid classical stochastic process.

John R. Klauder

Piotr Garbaczewski, Robert Olkiewicz

In the paper with the above-noted title, D. T. Gillespie [Phys. Rev. A 49, 1607 (1994)] claims that the theory of Markov stochastic processes cannot provide an adequate mathematical framework for quantum mechanics. In conjunction with the specific quantum dynamics considered there, we give a general analysis of the associated dichotomic jump processes. If we assume that Gillespie's ``measurement probabilities'' are the transition probabilities of a stochastic process, then the process must have an invariant (time independent) probability measure. Alternatively, if we demand the probability measure of the process follow the quantally implemented (via the Born statistical postulate) evolution, then we arrive at the jump process which can be interpreted as a Markov process if restricted to a suitable duration time. However, there is no corresponding Markov process consistent with the ${\mathit{Z}}_{2}$ event space assumption, if we require its existence for all times t\ensuremath{\in}${\mathbf{R}}_{+}$. \textcopyright{} 1996 The American Physical Society.