Quantum processes

A number of ideas and questions related to the construction of quantum processes are discussed. Quantum state extension, entanglement and asymptotic behaviour of the entropy are some of the issues explored. These topics are studied in more detail for a class of quantum processes known as finitely correlated states. Several examples of such processes are presented, specifically a Free Fermionic model.Comment: 20 pages, 2 figures, to appear in the proceedings of the 46th Karpacz Winter School of Theoretical Physics "Quantum Dynamics and Information: Theory and Experiment

Exchangeable coalescents, ultrametric spaces, nested interval-partitions: A unifying approach

Kingman (1978)'s representation theorem states that any exchangeable partition of $\mathbb{N}$ can be represented as a paintbox based on a random mass-partition. Similarly, any exchangeable composition (i.e. ordered partition of $\mathbb{N}$) can be represented as a paintbox based on an interval-partition (Gnedin 1997). Our first main result is that any exchangeable coalescent process (not necessarily Markovian) can be represented as a paintbox based on a random non-decreasing process valued in interval-partitions, called nested interval-partition, generalizing the notion of comb metric space introduced by Lambert & Uribe Bravo (2017) to represent compact ultrametric spaces. As a special case, we show that any $\Lambda$-coalescent can be obtained from a paintbox based on a unique random nested interval partition called $\Lambda$-comb, which is Markovian with explicit transitions. This nested interval-partition directly relates to the flow of bridges of Bertoin & Le Gall (2003). We also display a particularly simple description of the so-called evolving coalescent (Pfaffelhuber & Wakolbinger 2006) by a comb-valued Markov process. Next, we prove that any measured ultrametric space $U$, under mild measure-theoretic assumptions on $U$, is the leaf set of a tree composed of a separable subtree called the backbone, on which are grafted additional subtrees, which act as star-trees from the standpoint of sampling. Displaying this so-called weak isometry requires us to extend the Gromov-weak topology of Greven et al (2006), that was initially designed for separable metric spaces, to non-separable ultrametric spaces. It allows us to show that for any such ultrametric space $U$, there is a nested interval-partition which is 1) indistinguishable from $U$ in the Gromov-weak topology; 2) weakly isometric to $U$ if $U$ has complete backbone; 3) isometric to $U$ if $U$ is complete and separable.Comment: 62 pages, 7 figure

General Framework for the Behaviour of Continuously Observed Open Quantum Systems

We develop the general quantum stochastic approach to the description of quantum measurements continuous in time. The framework, that we introduce, encompasses the various particular models for continuous-time measurements condsidered previously in the physical and the mathematical literature.Comment: 30 pages, no figure

Intrinsic randomness of unstable dynamics and Sz.-Nagy-Foias dilation theory

Misra, Prigogine and Courbage (MPC) demonstrated the possibility of obtaining stochastic Markov processes from deterministic dynamics simply through a "change of representation" which involves no loss of information provided the dynamical system under consideration has a suitably high degree of instability of motion. From a mathematical point of view, MPC theory is a theory of positivity preserving quasi-affine transformations that intertwine the unitary groups associated with deterministic dynamics to contraction semigroups associated with stochastic Markov processes. In this work, dropping the positivity condition, a characterization of the contraction semigroups induced by quasi-affine transformations, the structure of the unitary groups admitting such intertwining relations and a prototype for the quasi-affinities are given on the basis of the Sz.-Nagy-Foia\c{s} dilation theory. The results are applied to MPC theory in the context of statistical mechanics.Comment: 25 page

A Remark on the structure of symmetric quantum dynamical semigroups on von Neumann algebras

We study the structure of the generator of a symmetric, conservative quantum dynamical semigroup with norm-bounded generator on a von Neumann algebra equipped with a faithful semifinite trace. For von Neumann algebras with abelian commutant (i.e. type I von Neumann algebras), we give a necessary and sufficient algebraic condition for the generator of such a semigroup to be written as a sum of square of self-adjoint derivations of the von Neumann algebra. This generalizes some of the results obtained by Albeverio, H(phi)egh-Krohn and Olsen [Alb] for the special case of the finite dimensional matrix algebras. We also study similar questions for a class of quantum dynamical semigroups with unbounded generators.Comment: accepted in Infinite Dimensional Analysis, Quantum Probability and Related Toplics (World Scientific

Weak mixing properties for nonsingular actions

For a general group G we consider various weak mixing properties of nonsingular actions. In the case where the action is actually measure preserving all these properties coincide, and our purpose here is to check which implications persist in the nonsingular case.Comment: There is a lot of new stuff in this versio

A Knob for Markovianity

We study the Markovianity of a composite system and its subsystems. We show how the dissipative nature of a subsystem's dynamics can be modified without having to change properties of the composite system environment. By preparing different system initial states or dynamically manipulating the subsystem coupling, we find that it is possible to induce a transition from Markov to non-Markov behavior, and vice versa.Comment: V1: 5 pages, 2 figures. V2: extended version. 7 pages, 3 figures. A case of non-commuting system-environment interaction is discussed. Accepted as an FTC in NJ

Interaction representation method for Markov master equations in quantum optics

Conditions sufficient for a quantum dynamical semigroup (QDS) to be unital are proved for a class of problems in quantum optics with Hamiltonians which are self-adjoint polynomials of any finite order in creation and annihilation operators. The order of the Hamiltonian may be higher than the order of completely positive part of the formal generator of a QDS. The unital property of a minimal quantum dynamical semigroup implies the uniqueness of the solution of the corresponding Markov master equation in the class of quantum dynamical semigroups and, in the corresponding representation, it ensures preservation of the trace or unit operator. We recall that only in the unital case the formal generator of MME determines uniquely the corresponding QDS.Comment: 12 pages, LaTeX2e. To be published in: Trends in Mathematics, Stochastic Analysis and Mathematical Physics, ANESTOC, Proc. of the Fourth International Workshop, Birkhauser, Boston, 200

What does a generic Markov operator look like

We consider generic i.e., forming an everywhere dense massive subset classes of Markov operators in the space $L^2(X,\mu)$ with a finite continuous measure. Since there is a canonical correspondence that associates with each Markov operator a multivalued measure-preserving transformation (i.e., a polymorphism), as well as a stationary Markov chain, we can also speak about generic polymorphisms and generic Markov chains. The most important and inexpected generic properties of Markov operators (or Markov chains or polymorphisms) is nonmixing and totally nondeterministicity. It was not known even existence of such Markov operators (the first example due to M.Rozenblatt). We suppose that this class coinsided with the class of special random perturbations of $K$-automorphisms. This theory is measure theoretic counterpart of the theory of nonselfadjoint contractions and its application.Comment: 13 p.,Ref.1

On positive maps, entanglement and quantization

We outline the scheme for quantization of classical Banach space results associated with some prototypes of dynamical maps and describe the quantization of correlations as well. A relation between these two areas is discussed