16,465 research outputs found
Time-Minimal Control of Dissipative Two-level Quantum Systems: the Generic Case
The objective of this article is to complete preliminary results concerning
the time-minimal control of dissipative two-level quantum systems whose
dynamics is governed by Lindblad equations. The extremal system is described by
a 3D-Hamiltonian depending upon three parameters. We combine geometric
techniques with numerical simulations to deduce the optimal solutions.Comment: 24 pages, 16 figures. submitted to IEEE transactions on automatic
contro
Invariant control systems on Lie groups: A short survey
This is a short survey of our recent research on invariant control systems (and their associated optimal control problems). We are primarily concerned with equivalence and classification, especially in three dimensions.peerReviewe
Pricing options in illiquid markets: optimal systems, symmetry reductions and exact solutions
We study a class of nonlinear pricing models which involves the feedback
effect from the dynamic hedging strategies on the price of asset introduced by
Sircar and Papanicolaou. We are first to study the case of a nonlinear demand
function involved in the model. Using a Lie group analysis we investigate the
symmetry properties of these nonlinear diffusion equations. We provide the
optimal systems of subalgebras and the complete set of non-equivalent
reductions of studied PDEs to ODEs. In most cases we obtain families of exact
solutions or derive particular solutions to the equations.Comment: 14 page
Analytic Controllability of Time-Dependent Quantum Control Systems
The question of controllability is investigated for a quantum control system
in which the Hamiltonian operator components carry explicit time dependence
which is not under the control of an external agent. We consider the general
situation in which the state moves in an infinite-dimensional Hilbert space, a
drift term is present, and the operators driving the state evolution may be
unbounded. However, considerations are restricted by the assumption that there
exists an analytic domain, dense in the state space, on which solutions of the
controlled Schrodinger equation may be expressed globally in exponential form.
The issue of controllability then naturally focuses on the ability to steer the
quantum state on a finite-dimensional submanifold of the unit sphere in Hilbert
space -- and thus on analytic controllability. A relatively straightforward
strategy allows the extension of Lie-algebraic conditions for strong analytic
controllability derived earlier for the simpler, time-independent system in
which the drift Hamiltonian and the interaction Hamiltonia have no intrinsic
time dependence. Enlarging the state space by one dimension corresponding to
the time variable, we construct an augmented control system that can be treated
as time-independent. Methods developed by Kunita can then be implemented to
establish controllability conditions for the one-dimension-reduced system
defined by the original time-dependent Schrodinger control problem. The
applicability of the resulting theorem is illustrated with selected examples.Comment: 13 page
Description of Quantum Entanglement with Nilpotent Polynomials
We propose a general method for introducing extensive characteristics of
quantum entanglement. The method relies on polynomials of nilpotent raising
operators that create entangled states acting on a reference vacuum state. By
introducing the notion of tanglemeter, the logarithm of the state vector
represented in a special canonical form and expressed via polynomials of
nilpotent variables, we show how this description provides a simple criterion
for entanglement as well as a universal method for constructing the invariants
characterizing entanglement. We compare the existing measures and classes of
entanglement with those emerging from our approach. We derive the equation of
motion for the tanglemeter and, in representative examples of up to four-qubit
systems, show how the known classes appear in a natural way within our
framework. We extend our approach to qutrits and higher-dimensional systems,
and make contact with the recently introduced idea of generalized entanglement.
Possible future developments and applications of the method are discussed.Comment: 40 pages, 7 figures, 1 table, submitted for publication. v2: section
II.E has been changed and the Appendix on "Four qubit sl-entanglement
measure" has been removed. There are changes in the notation of section IV.
Typos and language mistakes has been corrected. A figure has been added and a
figure has been replaced. The references have been update
Invariant control systems and sub-Riemannian structures on lie groups: equivalence and isometries
In this thesis we consider invariant optimal control problems and invariant sub-Riemannian structures on Lie groups. Primarily, we are concerned with the equivalence and classification of problems (resp. structures). In the first chapter, both the class of invariant optimal control problems and the class of invariant sub-Riemannian structures are organised as categories. The latter category is shown to be functorially equivalent to a subcategory of the former category. Via the Pontryagin Maximum Principle, we associate to each invariant optimal control problem (resp. invariant sub-Riemannian structure) a quadratic Hamilton-Poisson system on the associated Lie-Poisson space. These Hamiltonian systems are also organised as a category and hence the Pontryagin lift is realised as a contravariant functor. Basic properties of these categories are investigated. The rest of this thesis is concerned with the classification (and investigation) of certain subclasses of structures and systems. In the second chapter, the homogeneous positive semidefinite quadratic Hamilton-Poisson systems on three-dimensional Lie-Poisson spaces are classified up to compatibility with a linear isomorphism; a list of 23 normal forms is exhibited. In the third chapter, we classify the invariant sub-Riemannian structures in three dimensions and calculate the isometry group for each normal form. By comparing our results with known results, we show that most isometries (in three dimensions) are in fact the composition of a left translation and a Lie group isomorphism. In the fourth and last chapter of this thesis, we classify the sub-Riemannian and Riemannian structures on the (2n + 1)-dimensional Heisenberg groups. Furthermore, we find the isometry group and geodesics of each normal form
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