3,967 research outputs found

    Maximal Commutative Subalgebras Invariant for CP-Maps: (Counter-)Examples

    Full text link
    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

    Get PDF
    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 L2L^2 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

    Full text link
    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

    Get PDF
    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 On+O_n^+ 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 On+O_n^+ and the twisted SUq(2)SU_q (2) quantum group.Comment: 54 pages, thoroughly revised, to appear in the Journal of Functional Analysi

    Quantum Markovian Subsystems: Invariance, Attractivity, and Control

    Full text link
    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

    Full text link
    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
    • …
    corecore