203 research outputs found
Relative "-Numerical Ranges for Applications in Quantum Control and Quantum Information
Motivated by applications in quantum information and quantum control, a new
type of "-numerical range, the relative "-numerical range denoted
, is introduced. It arises upon replacing the unitary group U(N) in
the definition of the classical "-numerical range by any of its compact and
connected subgroups .
The geometric properties of the relative "-numerical range are analysed in
detail. Counterexamples prove its geometry is more intricate than in the
classical case: e.g. is neither star-shaped nor simply-connected.
Yet, a well-known result on the rotational symmetry of the classical
"-numerical range extends to , as shown by a new approach based on
Lie theory. Furthermore, we concentrate on the subgroup , i.e. the -fold tensor product of SU(2),
which is of particular interest in applications. In this case, sufficient
conditions are derived for being a circular disc centered at
origin of the complex plane. Finally, the previous results are illustrated in
detail for .Comment: accompanying paper to math-ph/070103
Non-Existence of Positive Stationary Solutions for a Class of Semi-Linear PDEs with Random Coefficients
We consider a so-called random obstacle model for the motion of a
hypersurface through a field of random obstacles, driven by a constant driving
field. The resulting semi-linear parabolic PDE with random coefficients does
not admit a global nonnegative stationary solution, which implies that an
interface that was flat originally cannot get stationary. The absence of global
stationary solutions is shown by proving lower bounds on the growth of
stationary solutions on large domains with Dirichlet boundary conditions.
Difficulties arise because the random lower order part of the equation cannot
be bounded uniformly
Illustrating the Geometry of Coherently Controlled Quantum Channels
We extend standard Markovian open quantum systems (quantum channels) by
allowing for Hamiltonian controls and elucidate their geometry in terms of Lie
semigroups. For standard dissipative interactions with the environment and
different coherent controls, we particularly specify the tangent cones (Lie
wedges) of the respective Lie semigroups of quantum channels. These cones are
the counterpart of the infinitesimal generator of a single one-parameter
semigroup. They comprise all directions the underlying open quantum system can
be steered to and thus give insight into the geometry of controlled open
quantum dynamics. Such a differential characterisation is highly valuable for
approximating reachable sets of given initial quantum states in a plethora of
experimental implementations.Comment: condensed and updated version; 14 pages; comments welcom
The landscape of quantum transitions driven by single-qubit unitary transformations with implications for entanglement
This paper considers the control landscape of quantum transitions in
multi-qubit systems driven by unitary transformations with single-qubit
interaction terms. The two-qubit case is fully analyzed to reveal the features
of the landscape including the nature of the absolute maximum and minimum, the
saddle points and the absence of traps. The results permit calculating the
Schmidt state starting from an arbitrary two-qubit state following the local
gradient flow. The analysis of multi-qubit systems is more challenging, but the
generalized Schmidt states may also be located by following the local gradient
flow. Finally, we show the relation between the generalized Schmidt states and
the entanglement measure based on the Bures distance
Pulsating wave for mean curvature flow in inhomogeneous medium
We prove the existence and uniqueness of pulsating waves for the motion by mean curvature of an n-dimensional hypersurface in an inhomogeneous medium, represented by a periodic forcing. The main difficulty is caused by the degeneracy of the equation and the fact the forcing is allowed to change sign. Under the assumption of weak inhomogeneity, we obtain uniform oscillation and gradient bounds so that the evolving surface can be written as a graph over a reference hyperplane. The existence of an effective speed of propagation is established for any normal direction. We further prove the Lipschitz continuity of the speed with respect to the normal and various stability properties of the pulsating wave. The results are related to the homogenisation of mean curvature flow with forcing
Hydrodynamic limit of condensing two-species zero range processes with sub-critical initial profiles
Two-species condensing zero range processes (ZRPs) are interacting particle systems with two species of particles and zero range interaction exhibiting phase separation outside a domain of sub-critical densities. We prove the hydrodynamic limit of nearest neighbour mean zero two-species condensing ZRP with bounded local jump rate for sub-critical initial profiles, i.e., for initial profiles whose image is contained in the region of sub-critical densities. The proof is based on H.T. Yau’s relative entropy method, which relies on the existence of sufficiently regular solutions to the hydrodynamic equation. In the particular case of the species-blind ZRP, we prove that the solutions of the hydrodynamic equation exist globally in time and thus the hydrodynamic limit is valid for all times
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