3,967 research outputs found
Maximal Commutative Subalgebras Invariant for CP-Maps: (Counter-)Examples
We solve, mainly by counterexamples, many natural questions regarding maximal
commutative subalgebras invariant under CP-maps or semigroups of CP-maps on a
von Neumann algebra. In particular, we discuss the structure of the generators
of norm continuous semigroups on B(G) leaving a maximal commutative subalgebra
invariant and show that there exists Markov CP-semigroups on M_d without
invariant maximal commutative subalgebras for any d>2.Comment: After the elemenitation in Version 2 of a false class of examples in
Version 1, we now provide also correct examples for unital CP-maps and Markov
semigroups on M_d for d>2 without invariant masa
Periodic homogenization of non-local operators with a convolution type kernel
The paper deals with homogenization problem for a non-local linear operator
with a kernel of convolution type in a medium with a periodic structure. We
consider the natural diffusive scaling of this operator and study the limit
behaviour of the rescaled operators as the scaling parameter tends to 0. More
precisely we show that in the topology of resolvent convergence the family of
rescaled operators converges to a second order elliptic operator with constant
coefficients. We also prove the convergence of the corresponding semigroups
both in space and the space of continuous functions, and show that for
the related family of Markov processes the invariance principle holds
Resonances, Unstable Systems and Irreversibility: Matter Meets Mind
The fundamental time-reversal invariance of dynamical systems can be broken
in various ways. One way is based on the presence of resonances and their
interactions giving rise to unstable dynamical systems, leading to well-defined
time arrows. Associated with these time arrows are semigroups bearing time
orientations. Usually, when time symmetry is broken, two time-oriented
semigroups result, one directed toward the future and one directed toward the
past. If time-reversed states and evolutions are excluded due to resonances,
then the status of these states and their associated backwards-in-time oriented
semigroups is open to question. One possible role for these latter states and
semigroups is as an abstract representation of mental systems as opposed to
material systems. The beginnings of this interpretation will be sketched.Comment: 9 pages. Presented at the CFIF Workshop on TimeAsymmetric Quantum
Theory: The Theory of Resonances, 23-26 July 2003, Instituto Superior
Tecnico, Lisbon, Portugal; and at the Quantum Structures Association Meeting,
7-22 July 2004, University of Denver. Accepted for publication in the
Internation Journal of Theoretical Physic
Symmetries of L\'evy processes on compact quantum groups, their Markov semigroups and potential theory
Strongly continuous semigroups of unital completely positive maps (i.e.
quantum Markov semigroups or quantum dynamical semigroups) on compact quantum
groups are studied. We show that quantum Markov semigroups on the universal or
reduced C-algebra of a compact quantum group which are translation
invariant (w.r.t. to the coproduct) are in one-to-one correspondence with
L\'evy processes on its -Hopf algebra. We use the theory of L\'evy processes
on involutive bialgebras to characterize symmetry properties of the associated
quantum Markov semigroup. It turns out that the quantum Markov semigroup is
GNS-symmetric (resp. KMS-symmetric) if and only if the generating functional of
the L\'evy process is invariant under the antipode (resp. the unitary
antipode). Furthermore, we study L\'evy processes whose marginal states are
invariant under the adjoint action. In particular, we give a complete
description of generating functionals on the free orthogonal quantum group
that are invariant under the adjoint action. Finally, some aspects of
the potential theory are investigated. We describe how the Dirichlet form and a
derivation can be recovered from a quantum Markov semigroup and its L\'evy
process and we show how, under the assumption of GNS-symmetry and using the
associated Sch\"urmann triple, this gives rise to spectral triples. We discuss
in details how the above results apply to compact groups, group C-algebras
of countable discrete groups, free orthogonal quantum groups and the
twisted quantum group.Comment: 54 pages, thoroughly revised, to appear in the Journal of Functional
Analysi
Quantum Markovian Subsystems: Invariance, Attractivity, and Control
We characterize the dynamical behavior of continuous-time, Markovian quantum
systems with respect to a subsystem of interest. Markovian dynamics describes a
wide class of open quantum systems of relevance to quantum information
processing, subsystem encodings offering a general pathway to faithfully
represent quantum information. We provide explicit linear-algebraic
characterizations of the notion of invariant and noiseless subsystem for
Markovian master equations, under different robustness assumptions for
model-parameter and initial-state variations. The stronger concept of an
attractive quantum subsystem is introduced, and sufficient existence conditions
are identified based on Lyapunov's stability techniques. As a main control
application, we address the potential of output-feedback Markovian control
strategies for quantum pure state-stabilization and noiseless-subspace
generation. In particular, explicit results for the synthesis of stabilizing
semigroups and noiseless subspaces in finite-dimensional Markovian systems are
obtained.Comment: 16 pages, no figures. Revised version with new title, corrected
typos, partial rewriting of Section III.E and some other minor change
A hierarchical structure of transformation semigroups with applications to probability limit measures
The structure of transformation semigroups on a finite set is analyzed by
introducing a hierarchy of functions mapping subsets to subsets. The resulting
hierarchy of semigroups has a corresponding hierarchy of minimal ideals, or
kernels. This kernel hierarchy produces a set of tools that provides direct
access to computations of interest in probability limit theorems; in
particular, finding certain factors of idempotent limit measures. In addition,
when considering transformation semigroups that arise naturally from edge
colorings of directed graphs, as in the road-coloring problem, the hierarchy
produces simple techniques to determine the rank of the kernel and to decide
when a given kernel is a right group. In particular, it is shown that all
kernels of rank one less than the number of vertices must be right groups and
their structure for the case of two generators is described.Comment: 35 pages, 4 figure
- …