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

Journal of Physics A Mathematical and General

Quantum state diffusion, localization and quantum dispersion entropy

Nicolas Gisin, Ian Colin Percival

Published May 7, 1993• 191 citations
Publisher

Abstract

The quantum state diffusion model introduced in an earlier paper represents the evolution of an individual open quantum system by an Ito diffusion equation for its quantum state. The diffusion and drift terms in this equation are derived from interaction with the environment. In this paper two localization theorems are proved. The dispersion entropy theorem shows that under special conditions, which are commonly satisfied to a good approximation, the mean quantum dispersion entropy, which measures the mean dispersion or delocalization of the quantum states, decreases at a rate equal to a weighted sum of effective interaction rates, so that the localization always increases in the mean, except when the effective interaction with the environment is zero. The general localization theorem provides a formula for more general conditions.

DOI
10.1088/0305-4470/26/9/018

Related papers

Localization of wide-open quantum systems

Ian Colin Percival

In a wide-open quantum system, the effect of the Hamiltonian is negligible by comparison with the effect of the environment. For open systems, this is the opposite limit to closed or isolated systems. The quantum state diffusion model provides equations for the localization or reduction of quantum states of wide-open systems. The ensemble localization of an operator is defined, and it is proved that the rate of selflocalization of a selfadjoint environment operator towards one of its eigenstates is no less than 2. A bound is also obtained for the rate of selflocalization of some non-selfadjoint operators, which localize to minimum indeterminacy wave packets. The theory is presented for quasiclassical systems. For a sufficient number of independent environment operators, the states localize asymptotically to wave packets with Heisenberg indeterminacy products close to the minimum, which look to classical eyes like phase space points. To zeroth order in h(cross), the time-dependent WKB theory of quantum state diffusion due to a single operator shows localization or reduction within and between fixed classical sheets or Lagrangian manifolds. To first order, the sheets themselves diffuse. The rate of localization in an ensemble is determined by commutation terms with either sign and by correlation terms which always increase the localization. For the quasiclassical case the latter dominate, and this leads to a purely classical theory of localization, with a picture based on the diffusion of phase space densities. This means that state diffusion dynamics, like Hamiltonian dynamics, has a purely classical form, in which Planck's constant plays no role.

Journal of Physics A Mathematical and General•Feb 7, 1994•60 citations

The quantum state diffusion picture of physical processes

Nicolas Gisin, Ian Colin Percival

In earlier papers the authors introduced a quantum state diffusion model for the evolution of an individual open quantum system and proved localization theorems based on this model. This paper shows in more detail how the diffusion leads to localization in position and phase space, and to symmetry breaking for chiral molecules. The theory of radioactive decay of absorbers and detectors is described in the state diffusion picture. The Mott and Gurney theory of latent image formation in photography is presented in its state diffusion version. It is an example of quantum detection without significant amplification.

Journal of Physics A Mathematical and General•May 7, 1993•174 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

Quantum fluctuation theorem for dissipative processes

Gabriele De Chiara, Alberto Imparato

We present a general quantum fluctuation theorem for the entropy production of an open quantum system coupled to multiple environments, not necessarily at equilibrium. Such a general theorem, when restricted to the weak-coupling and Markovian regime, holds for both local and global master equations, corroborating the thermodynamic consistency of local quantum master equations. The theorem is genuinely quantum, as it can be expressed in terms of conservation of a Hermitian operator, describing the dynamics of the system state operator and of the entropy change in the baths. The integral fluctuation theorem follows from the properties of such an operator. Furthermore, it is also valid when the system is described by a time-dependent Hamiltonian. As such, the quantum Jarzynski equality is a particular case of the general result presented here. Moreover, our result can be extended to nonthermal baths, as long as microreversibility is preserved. We present some numerical examples to showcase the exact results previously obtained. We finally generalize the fluctuation theorem to the case where the interaction between the system and the bath is explicitly taken into account. We show that the fluctuation theorem amounts to a relation between time-reversed dynamics of the global density matrix and a two-time correlation function along the forward dynamics involving the baths' entropy alone.

arXiv:2108.05937•Physical Review Research•Jun 21, 2022•11 citations

A Generalized Entropy-Hamiltonian Relation in Open Quantum Systems

Hiroki Majima, Michio Tokuyama, Irwin Oppenheim +1 more

The state of a system changes as a consequence of its internal dynamics and of the interaction with the environment that leads to a system‐environment correlation and cannot be represented in terms of unitary Hamiltonian dynamics of the system. In order to investigate the role of interaction between the system and its environment, we derive a general formula which relates the Hamiltonian of the system to entropy. This formula enables us to discriminate a type of interactions which causes the change in the state of the system in a spontaneous process. In other words, this formula enables us to know the type of interactions which would induce a new equilibrium state of the system in the process. It is found that the certain type of interactions does not affect the state (dynamics) of the system even if there are interactions between the system and its environment.

AIP conference proceedings•Jan 1, 2008•2 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
The quantum state diffusion model introduced in an earlier paper represents the evolution of an individual open quantum system by an Ito diffusion equation for its quantum state. The diffusion and drift terms in this equation are derived from interaction with the environment. In this paper two localization theorems are proved. The dispersion entropy theorem shows that under special conditions, which are commonly satisfied to a good approximation, the mean quantum dispersion entropy, which measures the mean dispersion or delocalization of the quantum states, decreases at a rate equal to a weighted sum of effective interaction rates, so that the localization always increases in the mean, except when the effective interaction with the environment is zero. The general localization theorem provides a formula for more general conditions.
In a wide-open quantum system, the effect of the Hamiltonian is negligible by comparison with the effect of the environment. For open systems, this is the opposite limit to closed or isolated systems. The quantum state diffusion model provides equations for the localization or reduction of quantum states of wide-open systems. The ensemble localization of an operator is defined, and it is proved that the rate of selflocalization of a selfadjoint environment operator towards one of its eigenstates is no less than 2. A bound is also obtained for the rate of selflocalization of some non-selfadjoint operators, which localize to minimum indeterminacy wave packets. The theory is presented for quasiclassical systems. For a sufficient number of independent environment operators, the states localize asymptotically to wave packets with Heisenberg indeterminacy products close to the minimum, which look to classical eyes like phase space points. To zeroth order in h(cross), the time-dependent WKB theory of quantum state diffusion due to a single operator shows localization or reduction within and between fixed classical sheets or Lagrangian manifolds. To first order, the sheets themselves diffuse. The rate of localization in an ensemble is determined by commutation terms with either sign and by correlation terms which always increase the localization. For the quasiclassical case the latter dominate, and this leads to a purely classical theory of localization, with a picture based on the diffusion of phase space densities. This means that state diffusion dynamics, like Hamiltonian dynamics, has a purely classical form, in which Planck's constant plays no role.
In earlier papers the authors introduced a quantum state diffusion model for the evolution of an individual open quantum system and proved localization theorems based on this model. This paper shows in more detail how the diffusion leads to localization in position and phase space, and to symmetry breaking for chiral molecules. The theory of radioactive decay of absorbers and detectors is described in the state diffusion picture. The Mott and Gurney theory of latent image formation in photography is presented in its state diffusion version. It is an example of quantum detection without significant amplification.
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.
We present a general quantum fluctuation theorem for the entropy production of an open quantum system coupled to multiple environments, not necessarily at equilibrium. Such a general theorem, when restricted to the weak-coupling and Markovian regime, holds for both local and global master equations, corroborating the thermodynamic consistency of local quantum master equations. The theorem is genuinely quantum, as it can be expressed in terms of conservation of a Hermitian operator, describing the dynamics of the system state operator and of the entropy change in the baths. The integral fluctuation theorem follows from the properties of such an operator. Furthermore, it is also valid when the system is described by a time-dependent Hamiltonian. As such, the quantum Jarzynski equality is a particular case of the general result presented here. Moreover, our result can be extended to nonthermal baths, as long as microreversibility is preserved. We present some numerical examples to showcase the exact results previously obtained. We finally generalize the fluctuation theorem to the case where the interaction between the system and the bath is explicitly taken into account. We show that the fluctuation theorem amounts to a relation between time-reversed dynamics of the global density matrix and a two-time correlation function along the forward dynamics involving the baths' entropy alone.
The state of a system changes as a consequence of its internal dynamics and of the interaction with the environment that leads to a system‐environment correlation and cannot be represented in terms of unitary Hamiltonian dynamics of the system. In order to investigate the role of interaction between the system and its environment, we derive a general formula which relates the Hamiltonian of the system to entropy. This formula enables us to discriminate a type of interactions which causes the change in the state of the system in a spontaneous process. In other words, this formula enables us to know the type of interactions which would induce a new equilibrium state of the system in the process. It is found that the certain type of interactions does not affect the state (dynamics) of the system even if there are interactions between the system and its environment.
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.