1,264 research outputs found
An introduction to quantum filtering
This paper provides an introduction to quantum filtering theory. An
introduction to quantum probability theory is given, focusing on the spectral
theorem and the conditional expectation as a least squares estimate, and
culminating in the construction of Wiener and Poisson processes on the Fock
space. We describe the quantum It\^o calculus and its use in the modelling of
physical systems. We use both reference probability and innovations methods to
obtain quantum filtering equations for system-probe models from quantum optics.Comment: 41 pages, 1 figur
Stochastic Schrodinger equations as limit of discrete filtering
We consider an open model possessing a Markovian quantum stochastic limit and
derive the limit stochastic Schrodinger equations for the wave function
conditioned on indirect observations using only the von Neumann projection
postulate. We show that the diffusion (Gaussian) situation is universal as a
result of the central limit theorem with the quantum jump (Poissonian)
situation being an exceptional case. It is shown that, starting from the
correponding limiting open systems dynamics, the theory of quantum filtering
leads to the same equations, therefore establishing consistency of the quantum
stochastic approach for limiting Markovian models.Comment: 21 pages, no figure
Quantum filtering for multiple measurements driven by fields in single-photon states
In this paper, we derive the stochastic master equations for quantum systems
driven by a single-photon input state which is contaminated by quantum vacuum
noise. To improve estimation performance, quantum filters based on
multiple-channel measurements are designed. Two cases, namely diffusive plus
Poissonian measurements and two diffusive measurements, are considered.Comment: 8 pages, 6 figures, submitted for publication. Comments are welcome
The Separation Principle in Stochastic Control, Redux
Over the last 50 years a steady stream of accounts have been written on the
separation principle of stochastic control. Even in the context of the
linear-quadratic regulator in continuous time with Gaussian white noise, subtle
difficulties arise, unexpected by many, that are often overlooked. In this
paper we propose a new framework for establishing the separation principle.
This approach takes the viewpoint that stochastic systems are well-defined maps
between sample paths rather than stochastic processes per se and allows us to
extend the separation principle to systems driven by martingales with possible
jumps. While the approach is more in line with "real-life" engineering thinking
where signals travel around the feedback loop, it is unconventional from a
probabilistic point of view in that control laws for which the feedback
equations are satisfied almost surely, and not deterministically for every
sample path, are excluded.Comment: 23 pages, 6 figures, 2nd revision: added references, correction
Algebraic Structures and Stochastic Differential Equations driven by Levy processes
We construct an efficient integrator for stochastic differential systems
driven by Levy processes. An efficient integrator is a strong approximation
that is more accurate than the corresponding stochastic Taylor approximation,
to all orders and independent of the governing vector fields. This holds
provided the driving processes possess moments of all orders and the vector
fields are sufficiently smooth. Moreover the efficient integrator in question
is optimal within a broad class of perturbations for half-integer global root
mean-square orders of convergence. We obtain these results using the
quasi-shuffle algebra of multiple iterated integrals of independent Levy
processes.Comment: 41 pages, 11 figure
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