Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Stationary quantum Markov process for the Wigner function

Published Jan 1, 2007• 0 citations
OpenAlex

Abstract

Abstract. Many stochastic models have been investigated for quantum mechanics because of its stochastic nature. In 1988, Cohendet et al. introduced a dichotomic variable to quantum phase space and proposed a background Markov process for the time evolution of the Wigner function. However, in their method the whole distribution function is required to determine the next step of a constituent particle. In this paper, we discuss a stationary quantum Markov process which enables us to treat each particle independently introducing U(1) extension for the phase space. The process has branching and vanishing points but we can keep a finite time interval between the branchings. The procedure to make the simulation of the process is also discussed. Stationary quantum Markov process for the Wigner function 2 1.

Related papers

Stationary quantum Markov process for the Wigner function on a lattice phase space

T. Hashimoto, M. Horibe, A. Hayashi

As a stochastic model for quantum mechanics we present a stationary quantum Markov process for the time evolution of the Wigner function on a lattice phase space Z_N x Z_N with N odd. By introducing a phase factor extension to the phase space, each particle can be treated independently. This is an improvement on earlier methods that require the whole distribution function to determine the evolution of a constituent particle. The process has branching and vanishing points, though a finite time interval can be maintained between the branchings. The procedure to perform a simulation using the process is presented.

arXiv:quant-ph/0602227•Journal of Physics A Mathematical and Theoretical•Nov 6, 2007•5 citations

Quantum Markov Process on a Lattice

T. Hashimoto, M. Horibe, A. Hayashi

We develop a systematic description of Weyl and Fano operators on a lattice phase space. Introducing the so-called ghost variable even on an odd lattice, odd and even lattices can be treated in a symmetric way. The Wigner function is defined using these operators on the quantum phase space, which can be interpreted as a spin phase space. If we extend the space with a dichotomic variable, a positive distribution function can be defined on the new space. It is shown that there exits a quantum Markov process on the extended space which describes the time evolution of the distribution function. 1.

•Jan 1, 2003•0 citations

Fokker-Planck equation associated with the Wigner function of a quantum system with a finite number of states

O Cohendet, Ph. Combe, M. Sirugue-Collin

The authors consider a quantum system with a finite number N of states and show that a Markov process evolving in an 'extended' discrete phase space can be associated with the discrete Wigner function of the system. This Wigner function is built using the Weyl quantisation procedure on the group Z N *Z N . Moreover, the authors can use this process to compute the quantum mean values as probabilistic expectations of functions of this process. This probabilistic formulation can be seen as a stochastic mechanics in phase space.

Journal of Physics A Mathematical and General•Jun 7, 1990•10 citations

A computable branching process for the Wigner quantum dynamics

Sihong Shao, Yunfeng Xiong

A branching process treatment for the nonlocal Wigner pseudo-differential operator and its numerical applications in quantum dynamics is proposed and analyzed. We start from the discussion on two typical truncations of the nonlocal term, i.e., the k-truncated and y-truncated models. After introducing an auxiliary function γ(x), the (truncated) Wigner equation is reformulated into the integral formulation as well as its adjoint correspondence, both of which can be regarded as the renewal-type equations and have transparent stochastic interpretation. We prove that the moment of a branching process happens to be the solution for the adjoint equation, which connects rigorously the Wigner quantum dynamics to the stochastic branching process, and thus a sound mathematical framework for the Wigner Monte Carlo methods is established. Within the framework, the branching process for the y-truncated model recovers the popular signed particle Monte Carlo method which needs a discretization of the momentum space beforehand, whereas the k-truncated branching method allows a continuous momentum sampling. More interestingly, with the help of the rigorous connection, we find that a constant auxiliary function γ(x) ≡ γ0 performs better from the point of view of both theoretical and numerical aspects. In theory, we obtain a simple but analytical calculation formula for the growth rate of particle number, i.e., e2γ0t with t being the time. In numerics, the constant auxiliary function may facilitate the random sampling from the exponential distribution and improve the accuracy. Typical numerical experiments on the Gaussian barrier scattering validate our theoretical findings, as well as demonstrate the accuracy, the efficiency and thus the computability of the Wigner branching process.

•Mar 1, 2016•0 citations

Solution To Quantum Liouville Equation In Phase Space Via A Stochastic Process

Sarah John

A numerical technique for the solution of the Quantum Liouville equation is discussed which is expected to be useful for mixed states describing highly excited many body systems.;The equation is represented in phase-space, where the density operator becomes the Wigner function. The Liouville operator which determines its time evolution, separates into a classical and quantum term. When the Wigner function varies slowly in momentum, the quasi-classical approximation is valid in which one keeps only the first order quantum term in the Liouville equation.;The Wigner function is represented by a sample set of gaussian smeared points. In an iterative time development scheme, the points are propagated first along a classical phase-space trajectory, which is then followed by a quantum development. The latter is represented by a stochastic process incorporated into the numerical procedure as a generalized Markovian process.;This method is numerically tested for an anharmonic quartic potential. The contours of the Wigner function at different time intervals are compared with the purely classical and exact results. It is seen that it provides a definite improvement over the purely classical approximation. Various averages are also compared and sources of error discussed.;In addition it was discovered that intermediate mixed states show a 'quantum focusing' effect, wherein the function peaks to values greater than the original maximum. This quantum-mechanical effect did not appear in the pure case, and disappears because of Liouville's theorem for highly mixed states which are essentially classical. There are other numerical schemes, which treat the points as moving in an effective potential, wherein the points retain their identity. These again by Liouville's theorem cannot show this focusing effect.

•Jan 1, 1986•2 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
Abstract. Many stochastic models have been investigated for quantum mechanics because of its stochastic nature. In 1988, Cohendet et al. introduced a dichotomic variable to quantum phase space and proposed a background Markov process for the time evolution of the Wigner function. However, in their method the whole distribution function is required to determine the next step of a constituent particle. In this paper, we discuss a stationary quantum Markov process which enables us to treat each particle independently introducing U(1) extension for the phase space. The process has branching and vanishing points but we can keep a finite time interval between the branchings. The procedure to make the simulation of the process is also discussed. Stationary quantum Markov process for the Wigner function 2 1.
As a stochastic model for quantum mechanics we present a stationary quantum Markov process for the time evolution of the Wigner function on a lattice phase space Z_N x Z_N with N odd. By introducing a phase factor extension to the phase space, each particle can be treated independently. This is an improvement on earlier methods that require the whole distribution function to determine the evolution of a constituent particle. The process has branching and vanishing points, though a finite time interval can be maintained between the branchings. The procedure to perform a simulation using the process is presented.
We develop a systematic description of Weyl and Fano operators on a lattice phase space. Introducing the so-called ghost variable even on an odd lattice, odd and even lattices can be treated in a symmetric way. The Wigner function is defined using these operators on the quantum phase space, which can be interpreted as a spin phase space. If we extend the space with a dichotomic variable, a positive distribution function can be defined on the new space. It is shown that there exits a quantum Markov process on the extended space which describes the time evolution of the distribution function. 1.
The authors consider a quantum system with a finite number N of states and show that a Markov process evolving in an 'extended' discrete phase space can be associated with the discrete Wigner function of the system. This Wigner function is built using the Weyl quantisation procedure on the group Z N *Z N . Moreover, the authors can use this process to compute the quantum mean values as probabilistic expectations of functions of this process. This probabilistic formulation can be seen as a stochastic mechanics in phase space.
A branching process treatment for the nonlocal Wigner pseudo-differential operator and its numerical applications in quantum dynamics is proposed and analyzed. We start from the discussion on two typical truncations of the nonlocal term, i.e., the k-truncated and y-truncated models. After introducing an auxiliary function γ(x), the (truncated) Wigner equation is reformulated into the integral formulation as well as its adjoint correspondence, both of which can be regarded as the renewal-type equations and have transparent stochastic interpretation. We prove that the moment of a branching process happens to be the solution for the adjoint equation, which connects rigorously the Wigner quantum dynamics to the stochastic branching process, and thus a sound mathematical framework for the Wigner Monte Carlo methods is established. Within the framework, the branching process for the y-truncated model recovers the popular signed particle Monte Carlo method which needs a discretization of the momentum space beforehand, whereas the k-truncated branching method allows a continuous momentum sampling. More interestingly, with the help of the rigorous connection, we find that a constant auxiliary function γ(x) ≡ γ0 performs better from the point of view of both theoretical and numerical aspects. In theory, we obtain a simple but analytical calculation formula for the growth rate of particle number, i.e., e2γ0t with t being the time. In numerics, the constant auxiliary function may facilitate the random sampling from the exponential distribution and improve the accuracy. Typical numerical experiments on the Gaussian barrier scattering validate our theoretical findings, as well as demonstrate the accuracy, the efficiency and thus the computability of the Wigner branching process.
A numerical technique for the solution of the Quantum Liouville equation is discussed which is expected to be useful for mixed states describing highly excited many body systems.;The equation is represented in phase-space, where the density operator becomes the Wigner function. The Liouville operator which determines its time evolution, separates into a classical and quantum term. When the Wigner function varies slowly in momentum, the quasi-classical approximation is valid in which one keeps only the first order quantum term in the Liouville equation.;The Wigner function is represented by a sample set of gaussian smeared points. In an iterative time development scheme, the points are propagated first along a classical phase-space trajectory, which is then followed by a quantum development. The latter is represented by a stochastic process incorporated into the numerical procedure as a generalized Markovian process.;This method is numerically tested for an anharmonic quartic potential. The contours of the Wigner function at different time intervals are compared with the purely classical and exact results. It is seen that it provides a definite improvement over the purely classical approximation. Various averages are also compared and sources of error discussed.;In addition it was discovered that intermediate mixed states show a 'quantum focusing' effect, wherein the function peaks to values greater than the original maximum. This quantum-mechanical effect did not appear in the pure case, and disappears because of Liouville's theorem for highly mixed states which are essentially classical. There are other numerical schemes, which treat the points as moving in an effective potential, wherein the points retain their identity. These again by Liouville's theorem cannot show this focusing effect.
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.