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/0111139 · Journal of Physics A Mathematical and General

Exact positivity of the Wigner andP-functions of a Markovian open system

Lajos Diósi, Claus Kiefer

Published Mar 8, 2002• 46 citations
arXivHTMLPDF

Abstract

We discuss the case of a Markovian master equation for an open system, as it is frequently found from environmental decoherence. We prove two theorems for the evolution of the quantum state. The first one states that for a generic initial state the corresponding Wigner function becomes strictly positive after a finite time has elapsed. The second one states that also the P -function becomes exactly positive after a decoherence time of the same order. Therefore, the density matrix becomes exactly decomposable into a mixture of Gaussian pointer states.

DOI
10.1088/0305-4470/35/11/312

Related papers

Symplectic evolution of Wigner functions in Markovian open systems

Olivier Brodier, Alfredo M. Ozorio de Almeida

The Wigner function is known to evolve classically under the exclusive action of a quadratic Hamiltonian. If the system also interacts with the environment through Lindblad operators that are complex linear functions of position and momentum, then the general evolution is the convolution of a non-Hamiltonian classical propagation of the Wigner function with a phase space Gaussian that broadens in time. We analyze the consequences of this in the three generic cases of elliptic, hyperbolic, and parabolic Hamiltonians. The Wigner function always becomes positive in a definite time, which does not depend on the initial pure state. We observe the influence of classical dynamics and dissipation upon this threshold. We also derive an exact formula for the evolving linear entropy as the average of a narrowing Gaussian taken over a probability distribution that depends only on the initial state. This leads to a long time asymptotic formula for the growth of linear entropy. We finally discuss the possibility of recovering the initial state.

arXiv:quant-ph/0304087•Physical Review E•Jan 23, 2004•54 citations

Decoherence of semiclassical Wigner functions

Alfredo M. Ozorio de Almeida

The Lindblad master equation governs the general Markovian evolution of a density operator for an open quantum system. Semiclassical Wigner functions represent density operators in phase space in terms of chords on a classical manifold, so that the amplitude and phase of each chord contribution is classically defined. Inserting such a Wigner function into a phase space version of the master equation, its explicit evolution is derived in the absence of dissipation. There results a simple extension of the unitary evolution of the semiclassical Wigner function, which does not affect the phase of each chord contribution, while dampening its amplitude exponentially. Projecting the Wigner function on to an orthogonal position or momentum basis, the dampening of long chords emerges as the exponential decay of off-diagonal elements of the density matrix.

arXiv:quant-ph/0208094•Journal of Physics A Mathematical and General•Dec 12, 2002•13 citations

Time evolution of the Wigner function

Francisco Soto, P. Claverie

In this paper we give a partial answer to the problem: When does an initially non-negative Wigner function remain non-negative under the effect of the time evolution? We show that, for pure states, this is the case for linear systems only; to prove this we use the fact that the Wigner function is non-negative if and only if the wavefunction is Gaussian. We also prove that the Green’s solution of the evolution equation of the Wigner function, which in the framework of probability theory corresponds to the conditional probability density, takes on negative values. We utilize a theorem, about moments, borrowed from Pawula. We conclude that the Wigner phase-space formulation of quantum mechanics cannot receive a genuine probabilistic interpretation.

Journal of Mathematical Physics•May 1, 1983•21 citations

Non-Markovian master equations from piecewise dynamics

Bassano Vacchini

We construct a large class of completely positive and trace-preserving non-Markovian dynamical maps for an open quantum system. These maps arise from a piecewise dynamics for statistical operators characterized by a continuous time evolution interrupted by jumps, randomly distributed in time and described by a quantum channel. The state of the open system is shown to obey a closed evolution equation, given by a master equation with a memory kernel and an inhomogeneous term. The non-Markovianity of the obtained dynamics is explicitly assessed studying the behavior of the distinguishability of two different initial system's states with elapsing time.

arXiv:1301.2902•Physical Review A•Mar 8, 2013•61 citations

Wigner Centennial: His Function, and Its Environmental Decoherence

Lajos Diósi

In 1983, Wigner outlined a modified Schrödinger--von-Neumann equation of motion for macroobjects, to describe their typical coupling to the environment. This equation has become a principal model of environmental decoherence which is beleived responsible for the emergence ofclassicality in macroscopic quantum systems. Typically, this happens gradually and asymptotically after a certain characteristic decoherence time. For the Wigner-function, however, one can prove that it evolves perfectly into a classical (non-negative) phase space distribution after a finite time of decoherence.

arXiv:quant-ph/0212103•Dec 17, 2002•1 citation

Modeling the Observation of a Quantum System as an Interaction

Satish Ramakrishna

The act of measurement on a quantum state is supposed to collapse the state into one of several eigenstates of the operator corresponding to the observable being measured. This measurement process is sometimes described as outside standard quantum-mechanical evolution and not calculable from Schr\odinger's equation. There are two general approaches to the study of wave-function collapse: one called the consistent or decoherent histories approach and the other, the environmental decoherence approach, which studies the effect of the environment upon the quantum system, to explain wave-function collapse. In the environmental decoherence approach, one usually studies a Markovian-approximated Master equation to study the time-evolution of reduced density matrix and obtains the long-term dependence of the off-diagonal elements of this matrix. We do not make a Markovian assumption and study a particularly simple and calculable example. We find, the short-time behavior of a collapsing system, at least the one considered in this paper, is not exponential, which is a new result (the long-term behavior is, of course, still exponential). This allows one to connect the Fermi-golden rule quadratic-in-time behavior of a transition probability to the exponential long-time behavior of a collapsing wave-function.

•Nov 16, 2019•0 citations
We discuss the case of a Markovian master equation for an open system, as it is frequently found from environmental decoherence. We prove two theorems for the evolution of the quantum state. The first one states that for a generic initial state the corresponding Wigner function becomes strictly positive after a finite time has elapsed. The second one states that also the P -function becomes exactly positive after a decoherence time of the same order. Therefore, the density matrix becomes exactly decomposable into a mixture of Gaussian pointer states.
The Wigner function is known to evolve classically under the exclusive action of a quadratic Hamiltonian. If the system also interacts with the environment through Lindblad operators that are complex linear functions of position and momentum, then the general evolution is the convolution of a non-Hamiltonian classical propagation of the Wigner function with a phase space Gaussian that broadens in time. We analyze the consequences of this in the three generic cases of elliptic, hyperbolic, and parabolic Hamiltonians. The Wigner function always becomes positive in a definite time, which does not depend on the initial pure state. We observe the influence of classical dynamics and dissipation upon this threshold. We also derive an exact formula for the evolving linear entropy as the average of a narrowing Gaussian taken over a probability distribution that depends only on the initial state. This leads to a long time asymptotic formula for the growth of linear entropy. We finally discuss the possibility of recovering the initial state.
The Lindblad master equation governs the general Markovian evolution of a density operator for an open quantum system. Semiclassical Wigner functions represent density operators in phase space in terms of chords on a classical manifold, so that the amplitude and phase of each chord contribution is classically defined. Inserting such a Wigner function into a phase space version of the master equation, its explicit evolution is derived in the absence of dissipation. There results a simple extension of the unitary evolution of the semiclassical Wigner function, which does not affect the phase of each chord contribution, while dampening its amplitude exponentially. Projecting the Wigner function on to an orthogonal position or momentum basis, the dampening of long chords emerges as the exponential decay of off-diagonal elements of the density matrix.
In this paper we give a partial answer to the problem: When does an initially non-negative Wigner function remain non-negative under the effect of the time evolution? We show that, for pure states, this is the case for linear systems only; to prove this we use the fact that the Wigner function is non-negative if and only if the wavefunction is Gaussian. We also prove that the Green’s solution of the evolution equation of the Wigner function, which in the framework of probability theory corresponds to the conditional probability density, takes on negative values. We utilize a theorem, about moments, borrowed from Pawula. We conclude that the Wigner phase-space formulation of quantum mechanics cannot receive a genuine probabilistic interpretation.
We construct a large class of completely positive and trace-preserving non-Markovian dynamical maps for an open quantum system. These maps arise from a piecewise dynamics for statistical operators characterized by a continuous time evolution interrupted by jumps, randomly distributed in time and described by a quantum channel. The state of the open system is shown to obey a closed evolution equation, given by a master equation with a memory kernel and an inhomogeneous term. The non-Markovianity of the obtained dynamics is explicitly assessed studying the behavior of the distinguishability of two different initial system's states with elapsing time.
In 1983, Wigner outlined a modified Schrödinger--von-Neumann equation of motion for macroobjects, to describe their typical coupling to the environment. This equation has become a principal model of environmental decoherence which is beleived responsible for the emergence ofclassicality in macroscopic quantum systems. Typically, this happens gradually and asymptotically after a certain characteristic decoherence time. For the Wigner-function, however, one can prove that it evolves perfectly into a classical (non-negative) phase space distribution after a finite time of decoherence.
The act of measurement on a quantum state is supposed to collapse the state into one of several eigenstates of the operator corresponding to the observable being measured. This measurement process is sometimes described as outside standard quantum-mechanical evolution and not calculable from Schr\odinger's equation. There are two general approaches to the study of wave-function collapse: one called the consistent or decoherent histories approach and the other, the environmental decoherence approach, which studies the effect of the environment upon the quantum system, to explain wave-function collapse. In the environmental decoherence approach, one usually studies a Markovian-approximated Master equation to study the time-evolution of reduced density matrix and obtains the long-term dependence of the off-diagonal elements of this matrix. We do not make a Markovian assumption and study a particularly simple and calculable example. We find, the short-time behavior of a collapsing system, at least the one considered in this paper, is not exponential, which is a new result (the long-term behavior is, of course, still exponential). This allows one to connect the Fermi-golden rule quadratic-in-time behavior of a transition probability to the exponential long-time behavior of a collapsing wave-function.