Quantum Probability and Infinite Dimensional Analysis

INTRODUCTION

CONTENTS

Let μ be a probability measure on ℝ^d with finite moments of all orders. Then we can define the creation operator a⁺(j), the annihilation operator a^–(j), and the neutral operator a⁰(j) for each coordinate 1 ≤ j ≤ d. We use the neutral operators a⁰(i) and the commutators [a^–(j), a⁺(k)] to characterize polynomially symmetric, polynomially factorizable, and moment-equal probability measures. We also present some results for probability measures on the real line with finite support, infinite support, and compact support.

No abstract received.

We consider the (causally) normally ordered form of the quantum white noise equation for a Fermi white noise. We find a new form of the normally ordered equation for some class of Hamiltonians and we obtain the inner Langevin equation for such Hamiltonians.

Let μ be a probability measure on ℝ with finite moments of all orders. Suppose μ is not supported by a finite set of points. Then there exists a unique sequence of orthogonal polynomials such that P_n(x) is a polynomial of degree n with leading coefficient 1 and the equality (x − α_n)P_n(x) = P_n+1(x)+ω_nP_n−1(x) holds. The numbers are called the Jacobi-Szegö parameters of μ. The family determines the interacting Fock space of μ. In this paper we use the concept of generating function to give several methods for computing the orthogonal polynomials P_n(x) and the Jacobi-Szegö parameters α_n and ω_n. We also describe how to identify the orthogonal polynomials in terms of differential or difference operators.

In this paper we review some recent results concerning the analysis of the Open BCS model as proposed by Buffet and Martin as well as some generalizations. Our attention is mainly focused on the computation of the critical temperature.

No abstract received.

A class of vacuum-adapted regular quantum semimartingales, with integrands which act as the conditional expectation on Fock space, are proved to possess increments which are monotone independent. The vacuum-adapted analogue of the Poisson process is shown to have increments which are distributed according to the monotonic law of small numbers.

We use special chains of test functions and distributions spaces to give an analytic characterization theorem for continuous linear operators on these test spaces in terms of their symbols. Then, we define the notion of convolution of operators and we give explicit solutions of some quantum stochastic differential equations.

Normal *-homomorphic quantum stochastic cocycles on a von Neumann algebra determine *-endomorphism semigroups and thereby product systems of Hilbert W*-bimodules. This product system is identified for a wide class of stochastic cocycles including all those whose Markov semigroup is norm continuous. When the stochastic cocycle is viewed as a dilation of its Markov semigroup the dilation is typically not minimal however in many cases it dominates an irreducible quantum stochastic dilation. Moreover when the semigroup is norm continuous it has irreducible stochastic dilations and, if the semigroup is also cosnervative, the product system of any such dilation is the same as that of the minimal dilation.

Here we discuss interaction of a single two-level atom with a single mode of interacting electromagnetic field in the Jaynes-Cummings model with the rotating wave approximation.

A formulation of quantum mechanics on S¹ or its N-point discretisation based on different types of q-deformations of subalgebras of the kinematical algebra of the system was discussed ( [1] ) in the framework of Borel quantisation. We review this method and introduce new Hopf q-deformations of the full kinematical algebra, i.e. q-deformations of both the subalgebras of position and of momentum observable. The implications of this deformed approach to the dynamics and the resulting evolution equations is assessed and compared with previous results including the non-deformed case. The presented algebraic method for q-deformations can be translated to Lévy processes on algebraic structures and the related evolutions; possible applications are outlined.

Algebraic random walks (ARW) and quantum mechanical random walks (QRW) are investigated and related. Based on minimal data provided by the underlying bialgebras of functions defined on e. g. the real line R, the abelian finite group Z _N, and the canonical Heisenberg-Weyl algebra hw, and by introducing appropriate functionals on those algebras, examples of ARWs are constructed. These walks involve short and long range transition probabilities as in the case of R walk, bistochastic matrices as for the case of Z_N walk, or coherent state vectors as in the case of hw walk. The increase of classical entropy due to majorization order of those ARWs is shown, and further their corresponding evolution equations are obtained. Especially for the case of hw ARW, the diffusion limit of evolution equation leads to a quantum master equation for the density matrix of a boson system interacting with a bath of quantum oscillators prepared in squeezed vacuum state. A number of generalizations to other types of ARWs and some open problems are also stated. Next, QRWs are briefly presented together with some of their distinctive properties, such as their enhanced diffusion rates, and their behavior in respect to the relation of majorization to quantum entropy. Finally, the relation of ARWs to QRWs is investigated in terms of the theorem of unitary extension of completely positive trace preserving (CPTP) evolution maps by means of auxiliary vector spaces. It is applied to extend the CPTP step evolution map of a ARW for a quantum walker system into a unitary step evolution map for an associated QRW of a walker+quantum coin system. Examples and extensions are provided.

Starting from the multiplication table for a basis of a Lie algebra, we show how to find a realization as vector fields in the case there is a Lie flag. Then we show how this is connected to finding coordinates of the second kind. We find the explicit form for coordinates of the second kind for the Schrödinger algebra in a particularly useful basis. These coordinates are related to dual vector fields that form an abelian Lie algebra generating an associated family of polynomials. Families of polynomials in quantum variables arise by specialization of the parameters of the first kind corresponding to quantum observables.

No abstract received.

No abstract received.

In this paper we consider nearest neighbour models where the spin takes values in the set Φ = {η₁, η₂, …, η_q} and is assigned to the vertices of the Cayley tree Γ^k. The Hamiltonian is defined by some given λ-function. We find a condition for the function λ to determine the type of the von Neumann algebra generated by the GNS - construction associated with the quantum Markov state corresponding to the unordered phase of the λ-model. Also we give some physical applications of the obtained result.

In this paper we investigate quantum logical gates of Fredkin type. The information 0 and 1 will be encoded by coherent states on a general Fock space, the interaction between the inputs will be a general beam splitting procedure. The interaction in the control gate is given by a unitary operator. The aim of the paper is to give examples for quantum channels transmitting some information such that the gate will fulfill the truth table. For the output we get simple explicit expressions. Our aim is to construct a gate using the well-known splitting procedures and coherent states.

Various local hidden variables models for the singlet correlations exploit the detection loophole, or other loopholes connected with post-selection on coincident arrival times. I consider the connection with a probabilistic simulation technique called rejection-sampling, and pose some natural questions concerning what can be achieved and what cannot be achieved with local (or distributed) rejection sampling. In particular a new and more serious loophole, which we call the coincidence loophole, is introduced.

We analyse an argument of Deutsch, which purports to show that the deterministic part of classical quantum theory together with deterministic axioms of classical decision theory, together imply that a rational decision maker behaves as if the probabilistic part of quantum theory (Born’s law) is true. We uncover two missing assumptions in the argument, and show that the argument also works for an instrumentalist who is prepared to accept that the outcome of a quantum measurement is random in the frequentist sense: Born’s law is a consequence of functional and unitary invariance principles belonging to the deterministic part of quantum mechanics. Unfortunately, it turns out that after the necessary corrections we have done no more than give an easier proof of Gleason’s theorem under stronger assumptions. However, for some special cases the proof method gives positive results while using different assumptions to Gleason. This leads to the conjecture that the proof could be improved to give the same conclusion as Gleason under unitary invariance together with a much weaker functional invariance condition.

We consider whether the noncanonical Volterra representation may have an infinite-dimensional orthogonal complement or not by the use of the method of the stationary processes.

Ordered double product integrals such as are defined as formal power series with coefficients in the space of tensors over a Lie algebra in which the Lie bracket is got by taking commutators in an associative algebra, where r[h] is a formal power series with coefficients in and vanishing zero order term. They are characterised by quasitriangularity identities. A necessary and sufficient condition for such a product integral to satisfy the quantum Yang-Baxter equation is applied to the quantisation problem for Lie bialgebras.

We examine the main notions of noncommutative independence, namely tensor, free, boolean and monotone independence. We collect the results on unification of these notions from the point of view of reducing them to tensor independence. We also show how to reduce Λ-freeness to tensor independence and demonstrate that in a similar way one can construct analogous mixtures of tensor independence and boolean independence as well as tensor independence and monotone independence.

We review the recent results on the Jacobi field of a (real-valued) Lévy process defined on a Riemannian manifold. In the case where the Lévy process is neither Gaussian, nor Poisson, the corresponding Jacobi field acts in an extended Fock space. We also give a unitary equivalent representation of the Jacobi field in a usual Fock space. This representation is inspired by a result by Accardi, Franz, and Skeide¹.

No abstract received.

A Lévy process in a Lie group is a process that possesses independent and stationary multiplicative increments. The theory of such processes is not merely an extension of the theory of Lévy processes in Euclidean spaces. Because of the unique structures possessed by the non-commutative Lie groups, these processes exhibit certain interesting properties which are not present for their counterparts in Euclidean spaces. These properties reveal a deep connection between the behavior of the stochastic processes and the underlying structures of the Lie groups. In this article, we will provide an introduction to Lévy processes in Lie groups, and present some old and new results in this area. The topics include, besides some general theory, Lévy processes in compact Lie groups, invariant Markov processes in homogeneous spaces, limiting properties of Lévy processes in semi-simple Lie groups of noncompact type, and dynamical aspects of such processes. Most of these results have been or will be published elsewhere, therefore, little proof will be given. See also Applebaum¹ for some other results in this area not mentioned here.

No abstract received.

The aims of this paper are to introduce the notion of generalized quantum Markov states, which extends translation-invariant quantum Markov states and C*-finitely correlated states in the setting on a UHF algebra to those on an AF C*-system with a shift homomorphism, and to show the extendability of generalized quantum Markov states on gauge-invariant parts of UHF algebras.

The white noise approach to the investigation of the dynamics of a quantum test particle interacting with a dilute Bose gas is presented. In this approach one proves that the appropriate operators of the Bose gas converge, in the sense of correlators, to operators constructed from some quantum white noise. The limiting dynamics is described by a quantum white noise equation or an equivalent quantum stochastic equation driven by a quantum Poisson process. These equations are applied to derivation of a quantum Langevin equation and a linear Boltzmann equation for the reduced density matrix of the test particle. The first part of the paper is devoted to the approach which was developed by L. Accardi, I. Volovich and the author and uses the Fock-antiFock (or GNS) representation for the CCR algebra of the gas. In the second part the development of the approach to the derivation of the limiting equations directly in terms of the correlation functions, without use of the Fock-antiFock representation, is described. This simplifies the derivation and allows to express the strength of the quantum number process directly in terms of the one-particle S-matrix.

The random variable which takes its values uniformly on the set u(m) of n × n unitary matrices (i.e. its distribution is the Haar measure of ) is the so-called Haar unitary. If we truncate the last m – n last columns and m – n bottom rows then an m × m random matrix is obtained. K. Życzkowski and H-J. Sommers examined the properties of the truncated Haar unitary, and they computed the joint eigenvalue distribution of U_[m,n]. Since this is a contraction, all the eigenvalues lie on the unit disc. Now we consider the case m = 2n, and we prove that the large deviation principle holds for the empirical eigenvalue distribution of U_[2n,n] as n → ∞. We determine the rate function from the logarithmic energy and the joint eigenvalue distribution of U_[2n,n], and by using results from potential theory, we minimalize the rate function in order to get the limit distribution.

We are traditionally educated to consider noise a nuisance, hindering the transmission and detection of signals. Recently, however, the paradigm of stochastic resonance (SR) proved this assertion wrong: indeed, the appropriate amount of noise can facilitate signal detection in noisy environments. Due to its simplicity and robustness, SR has been implemented by mother nature on almost any scale, from celestial mechanics to ion channels. The present minireview outlines the basic mechanism underlying SR phenomena, and discusses recent quantum optical scenarios, in the deep quantum regime.

In the first part of the paper we describe the natural scheme for proving noncommutative individual ergodic theorems, generalize it for multiple sequences of measurable operators affiliated with a semifinite von Neumann algebra M, and apply it to theorems concerning unrestricted convergence of multiaverages. In the second part we prove convergence of ergodic averages induced by several maps satisfying specific recurrence relations, including so-called Multi Free Group Partial Sums. This is the multiindexed version of results obtained earlier jointly with V.I.Chilin and S.Litvinov.

CP-semigroups on C*–algebras lead to product systems of Hilbert modules and quantum Lévy processes have an associated CP-semigroup. In these notes we review our latest results about the relation of the Arveson system of the Lévy process and the product system of its associated CP-semigroup and discuss some examples how the generators and product systems of classical Markov semigroups are related to that of brownian motion which is both a Markov process and a Lévy process. All results were obtained in collaboration with one or more of the following people: Luigi Accardi, Franco Fagnola, Uwe Franz, Volkmar Liebscher, Michael Schürmann.

We describe three methods to determine the structure of (sufficiently continuous) representations of the algebra 𝔅^a(E) of all adjointable operators on a Hilbert -module E by operators on a Hilbert C–module. While the last and latest proof is simple and direct and new even for normal representations of 𝔅(H) (H some Hilbert space), the other ones are direct generalizations of the representation theory of 𝔅(H) (based on Arveson’s and on Bhat’s approaches to product systems of Hilbert spaces) and depend on technical conditions (for instance, existence of a unit vector or restriction to von Neumann algebras and von Neumann modules). We explain why for certain problems the more specific information available in the older approaches is more useful for the solution of the problem.

We consider the strongly continuous unitary one parameter group on L²(ℝ) given by

The Hamiltonian is given by the singular operator

on a Sobolev space. The subspace where Ĥ is non-singular is the domain of the Hamiltonian H and the Hamiltonian coincides there with Ĥ. The spectrum of H is the real line. We calculate the generalized eigenvectors.

Let be an exact sequence of exact C*-algebras. Let α ∈ Aut(A) leaving J invariant and induced by α. We study the question when and , where ht denotes the Voiculescu-Brown topological entropy. There is a close connection to the existence of equivariant completely positive lifts. We introduce local equivariant lifts and prove that they always exist for a stabilized action. It follows that and at least for such stabilized actions.