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

arXiv:quant-ph/9506039 · Journal of Physics A Mathematical and General

Quantum state diffusion, localization and computation

R Schack, T A Brun, Ian Colin Percival

Published Sep 21, 1995• 92 citations
arXivHTMLPDF

Abstract

Numerical simulation of individual open quantum systems has proven advantages over density operator computations. Quantum state diffusion with a moving basis (MQSD) provides a practical numerical simulation method which takes full advantage of the localization of quantum states into wavepackets occupying small regions of classical phase space. Following and extending the original proposal of Percival, Alber and Steimle (1995), we show that MQSD can provide a further gain over ordinary QSD and other quantum trajectory methods of many orders of magnitude in computational space and time. Because of these gains, it is even possible to calculate an open quantum system trajectory when the corresponding isolated system is intractable. MQSD is particularly advantageous where classical or semiclassical dynamics provides an adequate qualitative picture but is numerically inaccurate because of significant quantum effects. The principles are illustrated by computations for the quantum Duffing oscillator and for second-harmonic generation in quantum optics. Potential applications in atomic and molecular dynamics, quantum circuits and quantum computation are suggested.

DOI
10.1088/0305-4470/28/18/028

Related papers

Quantum state diffusion, localization and computation

Rüdiger Schack, Todd A. Brun, Ian Colin Percival

Numerical simulation of individual open quantum systems has proven advantages over density operator computations. Quantum state diffusion with a moving basis (MQSD) provides a practical numerical simulation method which takes full advantage of the localization of quantum states into wave packets occupying small regions of classical phase space. Following and extending the original proposal of Percival, Alber and Steimle, we show that MQSD can provide a further gain over ordinary QSD and other quantum trajectory methods of many orders of magnitude in computational space and time. Because of these gains, it is even possible to calculate an open quantum system trajectory when the corresponding isolated system is intractable. MQSD is particularly advantageous where classical or semiclassical dynamics provides an adequate qualitative picture but is numerically inaccurate because of significant quantum effects. The principles are illustrated by computations for the quantum Duffing oscillator and for second harmonic generation in quantum optics. Potential applications in atomic and molecular dynamics, quantum circuits and quantum computation are suggested. J Phys A QMW preprint Th–95–19

•Jan 1, 1995•58 citations

Mixed classical-quantal representation for open quantum systems

T. Steimle, G. Alber, Ian Colin Percival

For all but the simplest open quantum systems, quantum trajectory Monte Carlo methods, including quantum jump and quantum state diffusion (QSD) methods, have besides their intuitive insight into the measurement process a numerical advantage over direct solutions for the density matrix, especially where many degrees of freedom are involved. For QSD the trajectories are continuous, and often localize to small near-minimum uncertainty wave packets which follow approximately classical paths in phase space. The mixed representation discussed here takes advantage of this localization to reduce computing space and time by a further significant factor, using a quantum oscillator representation that follows a classical path. The classical part of this representation describes the time evolution of the expectation values of position and momentum in classical phase space, while the quantal part determines the degree of localization of the quantum mechanical state around this phase space point. The method can be applied whether or not the localization is produced by a measuring apparatus.

Journal of Physics A Mathematical and General•Sep 21, 1995•24 citations

Quantum-state diffusion with a moving basis: Computing quantum-optical spectra

Rüdiger Schack, Todd A. Brun, Ian Colin Percival

Quantum-state diffusion (QSD) as a tool for solving quantum-optical master equations by stochastic simulation can be made several orders of magnitude more efficient if states in Hilbert space are represented in a moving basis of excited coherent states. The large savings in computer memory and time are due to the localization property of the QSD equation. We show how the method can be used to compute spectra and give an application to second harmonic generation. \textcopyright{} 1996 The American Physical Society.

arXiv:atom-ph/9510001•Physical Review A•Apr 1, 1996•29 citations

Classical mechanics from quantum state diffusion - a phase-space approach

Walter T. Strunz, Ian Colin Percival

Quantum state diffusion (QSD) provides a natural unravelling of a mixed-state open quantum system into component pure states. We investigate the semiclassical limit of QSD in a phase-space approach using the Wigner function. As , QSD exhibits two very different dynamical regimes, depending on the volume of phase space covered by the quantum state. For large volumes there is a localization regime represented by classical nonlinear and nonlocal diffusion processes. For small volumes, comparable in size with a Planck cell, there is a wavepacket regime . Here, the centroid of the wavepacket follows a classical Langevin equation, obtained through the adiabatic elimination of the dynamics of the second-order moments of the wavepacket. The corresponding Fokker-Planck equation is identical to the one obtained from the classical limit of the original mixed-state dynamics. In the companion paper we present an axiomatic approach to a classical theory of quantum localization without using the underlying QSD theory.

Journal of Physics A Mathematical and General•Feb 20, 1998•31 citations

Classical dynamics of quantum localization

Ian Colin Percival, Walter T. Strunz

Quantum state diffusion provides a dynamics for the localization or reduction of a quantum-mechanical wavepacket during a measurement or similar physical process. The essence of this localization dynamics is captured in the classical theory presented here, which applies to the common situation where the localization takes place over macroscopic distances and the wave properties of the system are no longer relevant. It provides a picture of the localization process in classical phase space, and a practical aid for computations on open quantum systems. The theory is developed from classical Hamiltonian dynamics in phase space and the known properties of localization. As an example it is used to illustrate how absorption by a screen leads to quantum jumps. A derivation from quantum state diffusion using the Wigner function is given in the companion paper.

Journal of Physics A Mathematical and General•Feb 20, 1998•16 citations

Quantum state diffusion, density matrix diagonalization, and decoherent histories: A model

J. J. Halliwell, Andreas Zoupas

We analyze the quantum evolution of a particle moving in a potential in interaction with an environment of harmonic oscillators in a thermal state, using the quantum state diffusion (QSD) picture of Gisin and Percival. The QSD picture exploits mathematical connection between the usual Markovian master equation for the evolution of the density operator and a class of stochastic nonlinear Schr\"odinger equations (Ito equations) for a pure state \ensuremath{\Vert}\ensuremath{\psi}〉, and appears to supply a good description of individual systems and processes. We find approximate stationary solutions to the Ito equation (exact, for the case of quadratic potentials). The solutions are Gaussians, localized around a point in phase space undergoing classical Brownian motion. We show, for quadratic potentials, that every initial state approaches these stationary solutions in the long time limit. We recover the density operator corresponding to these solutions, and thus show, for this particular model, that the QSD picture effectively supplies a prescription for approximately diagonalizing the density operator in a basis of phase space localized states. We show that the rate of localization is related to the decoherence time, and also to the time scale on which thermal and quantum fluctuations become comparable. We use these results to exemplify the general connection between the QSD picture and the decoherent histories approach to quantum mechanics, discussed previously by Di\'osi, Gisin, Halliwell, and Percival. \textcopyright{} 1995 The American Physical Society.

arXiv:quant-ph/9503008•Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Dec 15, 1995•127 citations
Numerical simulation of individual open quantum systems has proven advantages over density operator computations. Quantum state diffusion with a moving basis (MQSD) provides a practical numerical simulation method which takes full advantage of the localization of quantum states into wavepackets occupying small regions of classical phase space. Following and extending the original proposal of Percival, Alber and Steimle (1995), we show that MQSD can provide a further gain over ordinary QSD and other quantum trajectory methods of many orders of magnitude in computational space and time. Because of these gains, it is even possible to calculate an open quantum system trajectory when the corresponding isolated system is intractable. MQSD is particularly advantageous where classical or semiclassical dynamics provides an adequate qualitative picture but is numerically inaccurate because of significant quantum effects. The principles are illustrated by computations for the quantum Duffing oscillator and for second-harmonic generation in quantum optics. Potential applications in atomic and molecular dynamics, quantum circuits and quantum computation are suggested.
Numerical simulation of individual open quantum systems has proven advantages over density operator computations. Quantum state diffusion with a moving basis (MQSD) provides a practical numerical simulation method which takes full advantage of the localization of quantum states into wave packets occupying small regions of classical phase space. Following and extending the original proposal of Percival, Alber and Steimle, we show that MQSD can provide a further gain over ordinary QSD and other quantum trajectory methods of many orders of magnitude in computational space and time. Because of these gains, it is even possible to calculate an open quantum system trajectory when the corresponding isolated system is intractable. MQSD is particularly advantageous where classical or semiclassical dynamics provides an adequate qualitative picture but is numerically inaccurate because of significant quantum effects. The principles are illustrated by computations for the quantum Duffing oscillator and for second harmonic generation in quantum optics. Potential applications in atomic and molecular dynamics, quantum circuits and quantum computation are suggested. J Phys A QMW preprint Th–95–19
For all but the simplest open quantum systems, quantum trajectory Monte Carlo methods, including quantum jump and quantum state diffusion (QSD) methods, have besides their intuitive insight into the measurement process a numerical advantage over direct solutions for the density matrix, especially where many degrees of freedom are involved. For QSD the trajectories are continuous, and often localize to small near-minimum uncertainty wave packets which follow approximately classical paths in phase space. The mixed representation discussed here takes advantage of this localization to reduce computing space and time by a further significant factor, using a quantum oscillator representation that follows a classical path. The classical part of this representation describes the time evolution of the expectation values of position and momentum in classical phase space, while the quantal part determines the degree of localization of the quantum mechanical state around this phase space point. The method can be applied whether or not the localization is produced by a measuring apparatus.
Quantum-state diffusion (QSD) as a tool for solving quantum-optical master equations by stochastic simulation can be made several orders of magnitude more efficient if states in Hilbert space are represented in a moving basis of excited coherent states. The large savings in computer memory and time are due to the localization property of the QSD equation. We show how the method can be used to compute spectra and give an application to second harmonic generation. \textcopyright{} 1996 The American Physical Society.
Quantum state diffusion (QSD) provides a natural unravelling of a mixed-state open quantum system into component pure states. We investigate the semiclassical limit of QSD in a phase-space approach using the Wigner function. As , QSD exhibits two very different dynamical regimes, depending on the volume of phase space covered by the quantum state. For large volumes there is a localization regime represented by classical nonlinear and nonlocal diffusion processes. For small volumes, comparable in size with a Planck cell, there is a wavepacket regime . Here, the centroid of the wavepacket follows a classical Langevin equation, obtained through the adiabatic elimination of the dynamics of the second-order moments of the wavepacket. The corresponding Fokker-Planck equation is identical to the one obtained from the classical limit of the original mixed-state dynamics. In the companion paper we present an axiomatic approach to a classical theory of quantum localization without using the underlying QSD theory.
Quantum state diffusion provides a dynamics for the localization or reduction of a quantum-mechanical wavepacket during a measurement or similar physical process. The essence of this localization dynamics is captured in the classical theory presented here, which applies to the common situation where the localization takes place over macroscopic distances and the wave properties of the system are no longer relevant. It provides a picture of the localization process in classical phase space, and a practical aid for computations on open quantum systems. The theory is developed from classical Hamiltonian dynamics in phase space and the known properties of localization. As an example it is used to illustrate how absorption by a screen leads to quantum jumps. A derivation from quantum state diffusion using the Wigner function is given in the companion paper.
We analyze the quantum evolution of a particle moving in a potential in interaction with an environment of harmonic oscillators in a thermal state, using the quantum state diffusion (QSD) picture of Gisin and Percival. The QSD picture exploits mathematical connection between the usual Markovian master equation for the evolution of the density operator and a class of stochastic nonlinear Schr\"odinger equations (Ito equations) for a pure state \ensuremath{\Vert}\ensuremath{\psi}〉, and appears to supply a good description of individual systems and processes. We find approximate stationary solutions to the Ito equation (exact, for the case of quadratic potentials). The solutions are Gaussians, localized around a point in phase space undergoing classical Brownian motion. We show, for quadratic potentials, that every initial state approaches these stationary solutions in the long time limit. We recover the density operator corresponding to these solutions, and thus show, for this particular model, that the QSD picture effectively supplies a prescription for approximately diagonalizing the density operator in a basis of phase space localized states. We show that the rate of localization is related to the decoherence time, and also to the time scale on which thermal and quantum fluctuations become comparable. We use these results to exemplify the general connection between the QSD picture and the decoherent histories approach to quantum mechanics, discussed previously by Di\'osi, Gisin, Halliwell, and Percival. \textcopyright{} 1995 The American Physical Society.