Quantum Probability and Infinite-Dimensional Analysis

Preface

Contents

We review recent developments in the theory of quantum Markov states on the standard Z^d–spin lattice. A Dobrushin theory for quantum Markov fields is proposed. In the one–dimensional case where the order plays a crucial role, the structure arising from a quantum Markov state is fully understood. In this situation we obtain a splitting of a Markov state into a classical part, and a purely quantum part. This result allows us to provide a reconstruction theorem for quantum Markov states on chains.

Collective operators for generic quantum systems with discrete spectrum are investigated. These operators, considered as operators in the entangled Fock space (space generated by action of collective creations on the vacuum) satisfy a particular kind of quantum Boltzmann (or free) commutational relations.

The stationary quantum stochastic process j is introduced as a *-homomorphism embedding an involutive graded algebra into a ring of (abelian) cohomologies of the one-parameter group α consisting of *-automorphisms of a certain operator algebra in a Hilbert space such that every x from K_i is translated into an additive i – α-cocycle j(x). It is shown that (noncommutative) multiplicative Markovian cocycle defines a perturbation of the stationary quantum stochastic process in the sense of such definition. The E₀-semigroup on the von Neumann algebra associated with the Markovian perturbation of K-flow j posseses the restriction , , which is conjugate to the flow of Powers shifts β associated with j. It yields for an analogue of the Wold decomposition for classical stochastic processes on completely nondeterministic and deterministic parts. The examples of quantum stationary stochastic processes on the algebras of canonical commutation, anticommutation and square of white noise relations are considered. In the model situation of the space L²(R) all Markovian cocycles of the group of shifts are described up to unitary equivalence of perturbations.

We review here some recent work by L. Accardi and the author, concerning an alternative explanation of the experimental results concerning the resistivity tensor for a (almost) two-dimensional electron gas (2DEG) using the perturbative approach provided by the so-called stochastic limit. We also discuss the extention of this approach to a gas of quons.

Let 𝔥 be a Hilbert space and let be the von Neumann algebra of all bounded operators on 𝔥. We characterise w*-continuous Quantum Markov Semigroups enjoying the Feller property with respect to the -algebra of compact operators i.e. such that is -invariant and is astrongly continuous semigroup on . When is the minimal Quantum Markov Semigroup associated with quadratic forms given by £(x)[v, u] = 〈Gv, xu〉 + Σ_ℓ 〈L_ℓ v, xL_ℓ u〉 + 〈v, xGu〉 with possibly unbounded operators G, L_ℓ we show that the Feller property with respect to holds under a summability condition on the . We also show that the quantum Ornstein-Uhlenbeck semigroup enjoys the Feller property with respect to a bigger -algebra including and functions of position and momentum operators.

No abstract received.

No abstract received.

We introduce the concept of an adapted isometry which is an operator-theoretic characterization of the time evolution of a stationary stochastic process adapted to a filtration. Using a product decomposition of an adapted isometry it is shown that prediction errors with respect to the filtration correspond to a sequence of completely positive maps. Asymptotic properties of this correspondence are studied. In a special case the computations can be simplified by stochastic matrices.

The notion of generalised Brownian motion is extended to multiple processes indexed by a set . For −1 ≤ q ≤ 1 the q-product of positive definite functions on pair partitions having the multiplicative property is defined, and shown to be a positive definite function on -coloured pair partitions. The resulting -indexed generalised Brownian motion interpolates between graded tensor product (q = −1), reduced free product (q = 0) and tensor product (q = 1) of the given Brownian motions.

No abstract received.

We present a notion of quantum Markov random field based on a concept of conditional independence replacing the (usual) requirement of conditional expectation onto a desired algebra by conditional expectation onto a subalgebra, adopted to the special case that all subalgebras of the filtration are type I factors. It is immediate to introduce the classical notions like pairwise, local, global and factorizing Markov properties [11]. These share the same relations as in the classical case except the Hammersley-Clifford theorem, which remains open in the quantum case.

We provide in the context of quantum Markov chains due to ACCARDI coming up from iterated beam splittings a limit theorem concerning with a refinement of the discrete time. The limit object refers to a continuous time and should be called quantum Markov process. Closely related to our construction of such quantum Markov processes are isometric cocycles. We derive, upto technical conditions, a characterization of quasifree solutions of the cocycle equation.

Multiplicativity for Fock-adapted regular Markovian cocycles is proved for two cases, the first of which is new and facilitates dilation of quantum dynamical semi-groups on a separable -algebra. The proof of multiplicativity uses complete boundedness of the stochastic generator and the cocycle, induced maps on matrix spaces, and a simple commutative diagram.

We show that the transformations of random functionals by time changes on Brownian motion can be expressed as the adjoints of generalized Fourier-Mehler transforms. The derivatives of one-parameter families of such transformations of Brownian functionals are computed using a weighted Gross Laplacian and a second quantized operator.

In Reference¹⁷ Voiculescu generalizes his notion of free independent random variables¹⁶ to the notion ℨ-free ℨ-random variables (free independence with amalgamation over ℨ; see also Speicher¹⁵). Surprisingly, an amalgamated version of Bose independence resists to be meaningful in full generality version (roughly speaking, because there is in general no tensor product of ℨ-random variables). In these short notes (a slightly revised version of the preprint¹⁰) we intend to do not much more than to propose a definition of Bose ℨ-independence at least for so-called centered ℨ-random variables and to present some examples including creators and annihilators on the symmetric Fock module¹¹ and some random variables fulfilling q-comutation relations. The crucial notion of centered ℨ-random variables relies on the notion of centered Hilbert modules¹¹ and it can be shown that every -random variable (G some Hilbert space) is of that type; see References^13,2. Meanwhile, we know that every central limit distribution of Bose ℨ-independent ℨ-random variables may be represented by creators and annihilators on some symmetric Fock module; see Reference¹².

The investigation of products systems of Hilbert modules as introduced by Bhat and Skeide⁶ has now reached a state where it seems appropriate to give a summary of what we know about the structure. After showing how product systems appear naturally in the theory of dilations of CP-semigroups, it is one of the goals of these notes to give a list of solved and open problems.

In contrast with Arveson¹, who starts his theory of product systems of Hilbert spaces (Arveson systems, for short) with a concise definition of measurability conditions (which are equivalent to similar continuity conditions), the theory of product systems of Hilbert modules (in the sense of Definition 3.1 below) developed so far works without such conditions. While the algebraic constructions which work in that framework behave nicely with respect to topological completions or closures at a fixed "time", we could show continuity results to launch a definition of continuous tensor product system (Definition 7.1) and to show that this definition, although sufficiently general to contain all reasonable cases, does not have the described defect. In the case of type I and type II systems we found already a way to formulate continuity conditions in a less intrinsic way. Theorem 7.5 shows that the new definition is compatible with these special cases. We are, finally, able to define what we understand by a (continuous) type III product system, thus completing the classification scheme from Bhat and Skeide⁶, Barreto, Bhat, Liebscher and Skeide³ and Skeide¹⁹.

Quantum analogues of planar stochastic integrals of the first and second kind, as introduced by Wong and Zakai, are constructed for a quasi-free theory of fermions.

Antilinearity is quite natural in bipartite quantum systems: There is a one-to-one correspondence between vectors and certain antilinear maps, here called EPR-maps. Some of their properties and uses, including the factorization of quantum teleportation maps, is explained. There is an elementary link to twisted Kronecker products and to the modular objects of Tomita and Takesaki.