17,432 research outputs found
An Exponential Quantum Projection Filter for Open Quantum Systems
An approximate exponential quantum projection filtering scheme is developed
for a class of open quantum systems described by Hudson- Parthasarathy quantum
stochastic differential equations, aiming to reduce the computational burden
associated with online calculation of the quantum filter. By using a
differential geometric approach, the quantum trajectory is constrained in a
finite-dimensional differentiable manifold consisting of an unnormalized
exponential family of quantum density operators, and an exponential quantum
projection filter is then formulated as a number of stochastic differential
equations satisfied by the finite-dimensional coordinate system of this
manifold. A convenient design of the differentiable manifold is also presented
through reduction of the local approximation errors, which yields a
simplification of the quantum projection filter equations. It is shown that the
computational cost can be significantly reduced by using the quantum projection
filter instead of the quantum filter. It is also shown that when the quantum
projection filtering approach is applied to a class of open quantum systems
that asymptotically converge to a pure state, the input-to-state stability of
the corresponding exponential quantum projection filter can be established.
Simulation results from an atomic ensemble system example are provided to
illustrate the performance of the projection filtering scheme. It is expected
that the proposed approach can be used in developing more efficient quantum
control methods
A Quantum Langevin Formulation of Risk-Sensitive Optimal Control
In this paper we formulate a risk-sensitive optimal control problem for
continuously monitored open quantum systems modelled by quantum Langevin
equations. The optimal controller is expressed in terms of a modified
conditional state, which we call a risk-sensitive state, that represents
measurement knowledge tempered by the control purpose. One of the two
components of the optimal controller is dynamic, a filter that computes the
risk-sensitive state.
The second component is an optimal control feedback function that is found by
solving the dynamic programming equation. The optimal controller can be
implemented using classical electronics.
The ideas are illustrated using an example of feedback control of a two-level
atom
Quantum projection filter for a highly nonlinear model in cavity QED
Both in classical and quantum stochastic control theory a major role is
played by the filtering equation, which recursively updates the information
state of the system under observation. Unfortunately, the theory is plagued by
infinite-dimensionality of the information state which severely limits its
practical applicability, except in a few select cases (e.g. the linear Gaussian
case.) One solution proposed in classical filtering theory is that of the
projection filter. In this scheme, the filter is constrained to evolve in a
finite-dimensional family of densities through orthogonal projection on the
tangent space with respect to the Fisher metric. Here we apply this approach to
the simple but highly nonlinear quantum model of optical phase bistability of a
stongly coupled two-level atom in an optical cavity. We observe near-optimal
performance of the quantum projection filter, demonstrating the utility of such
an approach.Comment: 19 pages, 6 figures. A version with high quality images can be found
at http://minty.caltech.edu/papers.ph
Procedures for Converting among Lindblad, Kraus and Matrix Representations of Quantum Dynamical Semigroups
Given an quantum dynamical semigroup expressed as an exponential
superoperator acting on a space of N-dimensional density operators, eigenvalue
methods are presented by which canonical Kraus and Lindblad operator sum
representations can be computed. These methods provide a mathematical basis on
which to develop novel algorithms for quantum process tomography, the
statistical estimation of superoperators and their generators, from a wide
variety of experimental data. Theoretical arguments and numerical simulations
are presented which imply that these algorithms will be quite robust in the
presence of random errors in the data.Comment: RevTeX4, 31 pages, no figures; v4 adds new introduction and a
numerical example illustrating the application of these results to Quantum
Process Tomograph
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