23 research outputs found
The C-Numerical Range in Infinite Dimensions
In infinite dimensions and on the level of trace-class operators rather
than matrices, we show that the closure of the -numerical range is
always star-shaped with respect to the set , where
denotes the essential numerical range of the bounded operator .
Moreover, the closure of is convex if either is normal with
collinear eigenvalues or if is essentially self-adjoint. In the case of
compact normal operators, the -spectrum of is a subset of the
-numerical range, which itself is a subset of the convex hull of the closure
of the -spectrum. This convex hull coincides with the closure of the
-numerical range if, in addition, the eigenvalues of or are
collinear.Comment: 31 pages, no figures; to appear in Linear and Multilinear Algebr
Von Neumann Type of Trace Inequalities for Schatten-Class Operators
We generalize von Neumann's well-known trace inequality, as well as related
eigenvalue inequalities for hermitian matrices, to Schatten-class operators
between complex Hilbert spaces of infinite dimension. To this end, we exploit
some recent results on the -numerical range of Schatten-class operators. For
the readers' convenience, we sketched the proof of these results in the
Appendix.Comment: 16 page
The C-Numerical Range for Schatten-Class Operators
We generalize the -numerical range from trace-class to
Schatten-class operators, i.e. to and
with , and show that its closure
is always star-shaped with respect to the origin. For , this
is equivalent to saying that the closure of the image of the unitary orbit of
under any continous linear functional
is star-shaped with respect to the origin.
For , one has star-shapedness with respect to
, where denotes the essential range of
. Moreover, the closure of is convex if or is normal with
collinear eigenvalues. If and are both normal, then the -spectrum of
is a subset of the -numerical range, which itself is a subset of the
closure of the convex hull of the -spectrum. This closure coincides with the
closure of the -numerical range if, in addition, the eigenvalues of or
are collinear.Comment: 12 pages; extended version of the Addendum (linked in DOI) to a
previous article (arXiv:1712.01023
Exploring the Limits of Open Quantum Dynamics I: Motivation, New Results from Toy Models to Applications
Which quantum states can be reached by controlling open Markovian -level
quantum systems? Here, we address reachable sets of coherently controllable
quantum systems with switchable coupling to a thermal bath of temperature .
The core problem reduces to a toy model of studying points in the standard
simplex allowing for two types of controls: (i) permutations within the
simplex, (ii) contractions by a dissipative semigroup. By illustration, we put
the problem into context and show how toy-model solutions pertain to the
reachable set of the original controlled Markovian quantum system. Beyond the
case (amplitude damping) we present new results for using
methods of -majorisation.Comment: Extended abstract, accepted to conference MTNS 2020. The results by
d-majorisation are new and superseed arXiv:1905.01224 published as DOI:
10.1109/CDC40024.2019.9029452 . Part-II is arXiv:2003.04164 . arXiv admin
note: text overlap with arXiv:1905.0122
Exploring the Limits of Controlled Markovian Quantum Dynamics with Thermal Resources
Our aim is twofold: First, we rigorously analyse the generators of
quantum-dynamical semigroups of thermodynamic processes. We characterise a wide
class of GKSL-generators for quantum maps within thermal operations and argue
that every infinitesimal generator of (a one-parameter semigroup of) Markovian
thermal operations belongs to this class. We completely classify and visualise
them and their non-Markovian counterparts for the case of a single qubit.
Second, we use this description in the framework of bilinear control systems
to characterise reachable sets of coherently controllable quantum systems with
switchable coupling to a thermal bath. The core problem reduces to studying a
hybrid control system ("toy model") on the standard simplex allowing for two
types of evolution: (i) instantaneous permutations and (ii) a one-parameter
semigroup of -stochastic maps. We generalise upper bounds of the reachable
set of this toy model invoking new results on thermomajorisation. Using tools
of control theory we fully characterise these reachable sets as well as the set
of stabilisable states as exemplified by exact results in qutrit systems.Comment: 46 pages mai
Quantum optimal control in quantum technologies. Strategic report on current status, visions and goals for research in Europe
Quantum optimal control, a toolbox for devising and implementing the shapes of external fields that accomplish given tasks in the operation of a quantum device in the best way possible, has evolved into one of the cornerstones for enabling quantum technologies. The last few years have seen a rapid evolution and expansion of the field. We review here recent progress in our understanding of the controllability of open quantum systems and in the development and application of quantum control techniques to quantum technologies. We also address key challenges and sketch a roadmap for future developments
Quantum optimal control in quantum technologies. Strategic report on current status, visions and goals for research in Europe
Quantum optimal control, a toolbox for devising and implementing the shapes
of external fields that accomplish given tasks in the operation of a quantum
device in the best way possible, has evolved into one of the cornerstones for
enabling quantum technologies. The last few years have seen a rapid evolution
and expansion of the field. We review here recent progress in our understanding
of the controllability of open quantum systems and in the development and
application of quantum control techniques to quantum technologies. We also
address key challenges and sketch a roadmap for future developments.Comment: this is a living document - we welcome feedback and discussio