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Physical Review A

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

Lucién Hardy, Dipankar Home, Euan J. Squires, M. A. B. Whitaker

Published Oct 1, 1997• 5 citations
Publisher

Abstract

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.
DOI
10.1103/physreva.56.3301

Related papers

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.

Physical Review A•Aug 1, 1996•5 citations

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

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.

Physical Review A•Oct 1, 1997•3 citations

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.

arXiv:quant-ph/9601016•Physical Review A•Aug 1, 1996•4 citations

A Bohmian version of a 2-state quantum system

Matthew Dickau

I construct a Bohmian version of a 2-state quantum system, where in addition to the quantum state, the system is characterized by a definite physical state which changes stochastically under the guidance of the quantum state. I argue that the probabilities produced by this process are well-defined, contrary to claims in "Why quantum mechanics cannot be formulated as a Markov process" by D. Gillespie (1994) about a related model.

quant-pharXiv:2607.24480•Jul 28, 2026•0 citations

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

Daniel T. Gillespie

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.

Physical Review A•Jun 1, 1996•4 citations

Why quantum mechanics cannot be formulated as a Markov process

Daniel T. Gillespie

Both quantum mechanics and Markov-process theory deal with dynamical systems whose behaviors can be predicted only probabilistically. The question therefore arises as to whether Markov-process theory, a theory that is firmly rooted in the common-sense laws of probability, can provide a generally adequate mathematical framework for quantum mechanics, a theory whose predictions often seem to defy common sense. It is shown here that the measurable behavior of a certain elementary quantum system that oscillates between two discrete states cannot be modeled as a Markov process. That this will also be true for most other quantum systems is shown to follow from the same property of quantum mechanics that is responsible for the so-called quantum Zeno effect. The conclusion that quantum behavior cannot generally be modeled as a Markov process underscores the truly mysterious nature of quantum mechanics when viewed in common-sense terms.

Physical Review A•Mar 1, 1994•28 citations
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.
Both quantum mechanics and Markov-process theory deal with dynamical systems whose behaviors can be predicted only probabilistically. The question therefore arises as to whether Markov-process theory, a theory that is firmly rooted in the common-sense laws of probability, can provide a generally adequate mathematical framework for quantum mechanics, a theory whose predictions often seem to defy common sense. It is shown here that the measurable behavior of a certain elementary quantum system that oscillates between two discrete states cannot be modeled as a Markov process. That this will also be true for most other quantum systems is shown to follow from the same property of quantum mechanics that is responsible for the so-called quantum Zeno effect. The conclusion that quantum behavior cannot generally be modeled as a Markov process underscores the truly mysterious nature of quantum mechanics when viewed in common-sense terms.