scholarly journals THE STABILITY OF QUANTUM MARKOV FILTERS

2009 ◽  
Vol 12 (01) ◽  
pp. 153-172 ◽  
Author(s):  
RAMON VAN HANDEL
Keyword(s):  
Differential Equation ◽  
Stochastic Model ◽  
Stochastic Differential Equation ◽  
Initial State ◽  
Rank Condition ◽  
Quantum Stochastic Differential Equation ◽  
Absolutely Continuous ◽  
Finite Dimensional ◽  
Initial States ◽  
The Stability

When are quantum filters asymptotically independent of the initial state? We show that this is the case for absolutely continuous initial states when the quantum stochastic model satisfies an observability condition. When the initial system is finite dimensional, this condition can be verified explicitly in terms of a rank condition on the coefficients of the associated quantum stochastic differential equation.

scholarly journals Stochastic Model of the Absorption of Drugs Problem in the Cells and Organs

2021 ◽  
Vol 26 (1) ◽  
pp. 134-142
Author(s):  
Sajjad Abd-AL Hussein Haddad ALkha ◽  
Ihsan Khadim
Keyword(s):  
Differential Equation ◽  
Stochastic Model ◽  
Stochastic Differential Equation ◽  
Random Solution ◽  
Fixed Volume ◽  
The Stability

In this paper we create the stochastic differential equation for the problem of the Absorption of Drugs Problem in the Cells and Organs with fixed volume and then solve this equation and study the stability of the random solution.

scholarly journals Pure Gaussian state generation via dissipation: a quantum stochastic differential equation approach

2012 ◽  
Vol 370 (1979) ◽  
pp. 5324-5337 ◽  
Author(s):  
Naoki Yamamoto
Keyword(s):  
Differential Equation ◽  
Steady State ◽  
Stochastic Differential Equation ◽  
Cluster State ◽  
Dissipative System ◽  
Complete Characterization ◽  
Gaussian State ◽  
Quantum State Transfer ◽  
Quantum Stochastic Differential Equation ◽  
Gaussian System

Recently, the complete characterization of a general Gaussian dissipative system having a unique pure steady state was obtained. This result provides a clear guideline for engineering an environment such that the dissipative system has a desired pure steady state such as a cluster state. In this paper, we describe the system in terms of a quantum stochastic differential equation (QSDE) so that the environment channels can be explicitly dealt with. Then, a physical meaning of that characterization, which cannot be seen without the QSDE representation, is clarified; more specifically, the nullifier dynamics of any Gaussian system generating a unique pure steady state is passive. In addition, again based on the QSDE framework, we provide a general and practical method to implement a desired dissipative Gaussian system, which has a structure of quantum state transfer.

Quantum Stochastic Differential Equation is Unitarily Equivalent to a Symmetric Boundary Value Problem in Fock Space

1998 ◽  
Vol 01 (02) ◽  
pp. 175-199 ◽  
Author(s):  
Alexander M. Chebotarev
Keyword(s):  
Differential Equation ◽  
Schrödinger Equation ◽  
Boundary Value Problem ◽  
Stochastic Differential Equation ◽  
Schrodinger Equation ◽  
Fock Space ◽  
Boundary Value ◽  
Weak Limit ◽  
Quantum Stochastic Differential Equation ◽  
Markov Evolution

We show a new remarkable connection between the symmetric form of a quantum stochastic differential equation (QSDE) and the strong resolvent limit of the Schrödinger equations in Fock space: the strong resolvent limit is unitarily equivalent to QSDE in the adapted (or Ito) form, and the weak limit is unitarily equivalent to the symmetric (or Stratonovich) form of QSDE. We also prove that QSDE is unitarily equivalent to a symmetric boundary value problem for the Schrödinger equation in Fock space. The boundary condition describes standard jumps in phase and amplitude of components of Fock vectors belonging to the range of the resolvent. The corresponding Markov evolution equation (the Lindblad or Markov master equation) is derived from the boundary value problem for the Schrödinger equation.

On the theory and application of one-step numerical schemes for solving quantum stochastic differential equation (QSDE)

2014 ◽  
Vol 07 (03) ◽  
pp. 1450037
Author(s):  
T. O. Akinwumi ◽  
B. J. Adegboyegun
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Stochastic Differential Equation ◽  
Numerical Experiments ◽  
Numerical Schemes ◽  
Quantum Stochastic Differential Equation ◽  
One Step ◽  
Definition Of ◽  
Convergent Sequences ◽  
Solution Techniques

This paper presents one-step numerical schemes for solving quantum stochastic differential equation (QSDE). The algorithms are developed based on the definition of QSDE and the solution techniques yield rapidly convergent sequences which are readily computable. As well as developing the schemes, we perform some numerical experiments and the solutions obtained compete favorably with exact solutions. The solution techniques presented in this work can handle all class of QSDEs most especially when the exact solution does not exist.

scholarly journals Modelling based in the stochastic dynamics for the time evolution of the COVID-19

2020 ◽  
Author(s):  
Leonardo dos Santos Lima
Keyword(s):  
Differential Equation ◽  
Evolution Equation ◽  
Stochastic Model ◽  
Stochastic Differential Equation ◽  
Stochastic Dynamics ◽  
Planck Equation ◽  
Time Dynamics ◽  
Health Agencies ◽  
System Of Particles ◽  
Effective Description

Abstract The stochastic differential equation (SDE) corresponding to nonlinear Fokker-Planck equation where the nonlinearity appearing in this evolution equation can be interpreted as providing an effective description of a system of particles interacting is obtained. Additionally, we propose a stochastic model for time dynamics of the COVID-19 based in the set of data supported by the Brazilian health agencies.

ON THE HAMILTONIAN OPERATOR ASSOCIATED TO SOME QUANTUM STOCHASTIC DIFFERENTIAL EQUATIONS

2000 ◽  
Vol 03 (04) ◽  
pp. 483-503 ◽  
Author(s):  
M. GREGORATTI
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Unitary Group ◽  
Open Quantum System ◽  
Hamiltonian Operator ◽  
Stochastic Evolution ◽  
Quantum Stochastic Differential Equation ◽  
Special Cases ◽  
Densely Defined ◽  
Full Domain

We consider the quantum stochastic differential equation introduced by Hudson and Parthasarathy to describe the stochastic evolution of an open quantum system together with its environment. We study the (unbounded) Hamiltonian operator generating the unitary group connected, as shown by Frigerio and Maassen, to the solution of the equation. We find a densely defined restriction of the Hamiltonian operator; in some special cases we prove that this restriction is essentially self-adjoint and in one particular case we get the whole Hamiltonian with its full domain.

scholarly journals Distributional properties of solutions of dVt = Vt-dUt + dLt with Lévy noise

2011 ◽  
Vol 43 (3) ◽  
pp. 688-711 ◽  
Author(s):  
Anita Diana Behme
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Autocorrelation Function ◽  
Sufficient Conditions ◽  
Stationary Solutions ◽  
Necessary And Sufficient Conditions ◽  
Tail Behavior ◽  
Absolutely Continuous ◽  
Distributional Properties ◽  
Necessary And Sufficient

For a given bivariate Lévy process (Ut, Lt)t≥0, distributional properties of the stationary solutions of the stochastic differential equation dVt = Vt-dUt + dLt are analysed. In particular, the expectation and autocorrelation function are obtained in terms of the process (U, L) and in several cases of interest the tail behavior is described. In the case where U has jumps of size −1, necessary and sufficient conditions for the law of the solutions to be (absolutely) continuous are given.

scholarly journals Operations Stochastiques de Capitalisation

1986 ◽  
Vol 16 (S1) ◽  
pp. S5-S30 ◽  
Author(s):  
Pierre Devolder
Keyword(s):  
Differential Equation ◽  
Stochastic Model ◽  
Stochastic Differential Equation ◽  
Life Insurance ◽  
Conditional Expectation ◽  
Financial Risk ◽  
Numerical Examples ◽  
Premium Calculation

AbstractThis paper presents a stochastic model of capitalization which takes into account the financial risk in the actuarial processes.We first introduce a stochastic differential equation which allows us to define the capitalization and actualization processes.We use these concepts to present a new principle of premium calculation for the capitalization operations, based on the equality between backward reserve and conditional expectation of the forward reserve.A generalization of the classical Thiele equation in life insurance is also given.Numerical examples illustrate the model.

scholarly journals Simulation of Quantum Dynamics Based on the Quantum Stochastic Differential Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Ming Li
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Quantum Dynamics ◽  
Dynamical Behavior ◽  
Computational Cost ◽  
Quantum Master Equation ◽  
Level System ◽  
Quantum Stochastic Differential Equation ◽  
State Diffusion ◽  
Quantum State Diffusion

The quantum stochastic differential equation derived from the Lindblad form quantum master equation is investigated. The general formulation in terms of environment operators representing the quantum state diffusion is given. The numerical simulation algorithm of stochastic process of direct photodetection of a driven two-level system for the predictions of the dynamical behavior is proposed. The effectiveness and superiority of the algorithm are verified by the performance analysis of the accuracy and the computational cost in comparison with the classical Runge-Kutta algorithm.

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