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Physical Review A

Comment on ‘‘Why quantum dynamics can be formulated as a Markov process’’

Daniel T. Gillespie

Published Jun 1, 1996• 4 citations
Publisher

Abstract

In a paper with the title given above, Garbaczewski and Olkiewicz [Phys. Rev. A 51, 3445 (1995)] claim that a contrarily entitled paper of Gillespie [Phys. Rev. A 49, 1607 (1994)] is misleading because it is based on an allegedly overly restrictive assumption. In the discussion given here of that claim, the impugned assumption, and some foundational aspects of Markovian and non-Markovian stochastic processes, it is concluded that the thesis of the later title has not been established.

DOI
10.1103/physreva.53.4602

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Comment on ‘‘Why quantum mechanics cannot be formulated as a Markov process’’

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.

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Reply to ‘‘Comment on ‘Why quantum mechanics cannot be formulated as a Markov process’’’

Daniel T. Gillespie

In reply to the preceding Comment on an article by D. T. Gillespie [Phys. Rev. A 49, 1607 (1994)], it is emphasized that the most plausible interpretation of the singly conditioned probability function implies that the evolution of the two-state quantum system being considered cannot be characterized as a Markov process on those states over any time interval. And it is pointed out that an alternative interpretation that is offered in the Comment appears to be in conflict with quantum mechanics.

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Comment on “Why quantum mechanics cannot be formulated as a Markov process”

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We reply to suggestions made by Gillespie [Phys. Rev. A 49, 1607 (1994)] that our earlier paper [Phys. Rev. A 45, 4267 (1992)] contained mathematical inconsistencies. By use of a concrete model, we demonstrate explicitly that the behavior of a quantum-mechanical two-state oscillator may be simulated by a classical jump process.

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Reply to “Comment on ‘Why quantum mechanics cannot be formulated as a Markov process’ ”

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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.

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During the last ten years, the studies on non-Markovian open system dynamics has become increasingly popular and having contributions from diverse sets of research communities. This interest has arisen due to fundamental problematics as to how to define and quantify memory effects in the quantum domain, how to exploit and develop applications based on them, and also due to the following question: what are the ultimate limits for controlling open system dynamics? We give here a simple theoretical introduction to the basic approaches to define and quantify quantum non-Markovianity —also highlighting their connections and differences. In addition to the importance of the development for open quantum systems studies, we also discuss the implications of the progress for other fields including, e.g. , formal studies of stochastic processes and quantum information science, and conclude with possible future directions the recent developments open.

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In a paper with the title given above, Garbaczewski and Olkiewicz [Phys. Rev. A 51, 3445 (1995)] claim that a contrarily entitled paper of Gillespie [Phys. Rev. A 49, 1607 (1994)] is misleading because it is based on an allegedly overly restrictive assumption. In the discussion given here of that claim, the impugned assumption, and some foundational aspects of Markovian and non-Markovian stochastic processes, it is concluded that the thesis of the later title has not been established.
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 Z2{\mathit{Z}}_{2}Z2​ event space assumption, if we require its existence for all times t\ensuremath{\in}R+{\mathbf{R}}_{+}R+​. \textcopyright{} 1996 The American Physical Society.
During the last ten years, the studies on non-Markovian open system dynamics has become increasingly popular and having contributions from diverse sets of research communities. This interest has arisen due to fundamental problematics as to how to define and quantify memory effects in the quantum domain, how to exploit and develop applications based on them, and also due to the following question: what are the ultimate limits for controlling open system dynamics? We give here a simple theoretical introduction to the basic approaches to define and quantify quantum non-Markovianity —also highlighting their connections and differences. In addition to the importance of the development for open quantum systems studies, we also discuss the implications of the progress for other fields including, e.g. , formal studies of stochastic processes and quantum information science, and conclude with possible future directions the recent developments open.