5,125 research outputs found
Irreducible decompositions and stationary states of quantum channels
For a quantum channel (completely positive, trace-preserving map), we prove a
generalization to the infinite dimensional case of a result by Baumgartner and
Narnhofer. This result is, in a probabilistic language, a decomposition of a
general quantum channel into its irreducible positive recurrent components.
This decomposition is related with a communication relation on the reference
Hilbert space. This allows us to describe the full structure of invariant
states of a quantum channel, and of their supports
On the relationship between a quantum Markov semigroup and its representation via linear stochastic Schroedinger equations
A quantum Markov semigroup can be represented via classical diffusion
processes solving a stochastic Schr\"odinger equation. In this paper we first
prove that a quantum Markov semigroup is irreducible if and only if classical
diffusion processes are total in the Hilbert space of the system. Then we study
the relationship between irreducibility of a quantum Markov semigroup and
properties of these diffusions such as accessibility, the Lie algebra rank
condition, and irreducibility. We prove that all these properties are, in
general, weaker than irreducibility of the quantum Markov semigroup,
nevertheless, they are equivalent for some important classes of semigroups.Comment: 16 page
A Dynamical Approach to the Perron-Frobenius Theory and Generalized Krein-Rutman Type Theorems
We present a new dynamical approach to the classical Perron-Frobenius theory
by using some elementary knowledge on linear ODEs. It is completely
self-contained and significantly different from those in the literature. As a
result, we develop a complex version of the Perron-Frobenius theory and prove a
variety of generalized Krein-Rutman type theorems for real operators. In
particular, we establish some new Krein-Rutman type theorems for sectorial
operators in a formalism that can be directly applied to elliptic operators,
which allow us to reduce significantly the technical PDE arguments involved in
the study of the principal eigenvalue problems of these operators.Comment: 40 page
Geometry of free loci and factorization of noncommutative polynomials
The free singularity locus of a noncommutative polynomial f is defined to be
the sequence of hypersurfaces. The main
theorem of this article shows that f is irreducible if and only if is
eventually irreducible. A key step in the proof is an irreducibility result for
linear pencils. Apart from its consequences to factorization in a free algebra,
the paper also discusses its applications to invariant subspaces in
perturbation theory and linear matrix inequalities in real algebraic geometry.Comment: v2: 32 pages, includes a table of content
A strongly irreducible affine iterated function system with two invariant measures of maximal dimension
A classical theorem of Hutchinson asserts that if an iterated function system
acts on by similitudes and satisfies the open set condition then
it admits a unique self-similar measure with Hausdorff dimension equal to the
dimension of the attractor. In the class of measures on the attractor which
arise as the projections of shift-invariant measures on the coding space, this
self-similar measure is the unique measure of maximal dimension. In the context
of affine iterated function systems it is known that there may be multiple
shift-invariant measures of maximal dimension if the linear parts of the
affinities share a common invariant subspace, or more generally if they
preserve a finite union of proper subspaces of . In this note we
construct examples where multiple invariant measures of maximal dimension exist
even though the linear parts of the affinities do not preserve a finite union
of proper subspaces.Comment: This new version has a much more powerful version of the main theorem
and a less direct, more general approach to the proo
- …