Quantum Probability Communications

The following sections are included:

• CONTENTS

• PREFACE

We focus on some connections between classical and quantum stochastic calculus. It is shown how each calculus can be a great source of ideas and developments for the other.

We describe the construction of a quantum process with independent increments from a locally compact group G and a continuous conditionally negative definite function on G. We give two examples of such processes which exhibit intriguing connections with other parts of probability theory. The first example is concerned with the Heisenberg group and related to brownian paths, Ray-Knight theorems, and branching processes. The second one deals with automorphism group of trees and the free brownian motion.

This is an introductory article presenting some basic ideas of quantum probability. From a discussion of simple experiments with polarized light and a card game we deduce the necessity of extending the body of classical probability theory. For a class of systems, containing classical systems with finitely many states, a probabilistic model is developed. It can describe, in particular, the polarization experiments. Some examples of “quantum coin tosses” are discussed, closely related to V.F.R. Jones’ approach to braid group representations, to spin relaxation, and to nuclear magnetic resonance. In an appendix we indicate the steps which lead to the full mathematical model of quantum probability.

We review the mathematical formalism of the equilibrium quantum statistical mechanics of lattice models with an infinite degree of freedom. The equivalence of the KMS boundary condition, the Gibbs-Araki condition and the variational principle is established for a class of long range interactions. Our technical tool is one-parameter semigroups of completely positive maps on UHF C*-algebras, which we call Feller semigroups. Uniqueness of Gibbs and ground sates is re-examined from the viewpoint of unique ergodicity of Feller semigroups.

The following sections are included:

• INTRODUCTION

• S_n-ACTION IN TOY BOSON FOCK SPACE

• SOME REMARKS ON FINITE GROUP ACTIONS

• S_n ACTION IN THE FERMION FOCK SPACE

• References

The following sections are included:

• Introduction

• The categorial approach

• Monotone metrics on quantum states

• Observation levels without and with an priori state

• Bibliography

The following sections are included:

• Introduction

• The rotor model
  • Classical rotor
  • Quantum model

• Abstract quantum model

• Approximation

• References

We define the value, at any non-commutative finite stop time, of some vector semimartingales in Fock space. We apply it to stop a large class of operator processes, including all processes given by Hudson-Parthasarathy quantum stochastic integrals.

Every Evans-Hudson flow on the algebra of all bounded operators on a Hilbert space leads to a semigroup of *-endomorphisms, and then to a continuous tensor product system of Hilbert spaces. Here we have a new representation for exponential (Boson Fock space) product systems. This helps us to show that only such product systems arise from one-dimensional Evans-Hudson flows.

In this paper we develop an approach to defining quantum spectral integrals by exploiting the self–duality of a Hilbert space, and we obtain estimates for these integrals corresponding to those of P–A. Meyer and J–L. Journé for ordinary quantum stochastic integrals such as

The following sections are included:

• INTRODUCTION

• CONSTRUCTION

• FILTRATIONS

• SOLVING THE SYSTEM OF EQUATIONS IN §3G

• PARTIAL DEFINITION OF THE STOCHASTIC INTEGRAL

• LINEARITY

• REMOVING THE RESTRICTION ON THE SPECTRAL FAMILY

• THE VALUE IN THE ELEMENTARY CASE

• REFERENCES

The following sections are included:

• INTRODUCTION

• WAVE OPERATORS AND REDUCED DYNAMICS

• EXISTENCE OF THE WAVE OPERATORS

• AN EXAMPLE

• ACKNOWLEDGEMENTS

• REFERENCES

This paper concludes the study initiated in [GP] of orthogonally additive functionals on the full algebra B(H) of operators acting on a real or complex Hilbert space. The final result gives the general form of such a function. It can be viewed as a generalization of Gleason’s theorem, showing the possibility of extending a measure on projections to a (not necessarily linear) functional on the algebra which is additive on ‘orthogonal’ operators.

Long time evolution of the operator means is investigated for continuously observed finite dimensional quantum systems governed by the Belavkin quantum filtering equation. The coefficient of the quality of measurement is introduced and studied. This coefficient describes numerically the watch dog effect and the noncertainty principle for continuously observed quantum system under nondemolition measurement of diffusion type.

Reversely to the usual procedure, we construct multiple integrals with respect to an arbitrary 1–parametric (classical) white noise by the help of orthogonal polynomials. We get a chaos decomposition of the space of square integrable functionals of this white noise and an isomorphism between the white noise space and the symmetric Fock space with respect to the intensity measure of the noise. Moreover, the only noises which have multiple integrals in our, not very narrow, sense are characterized as Gaussian or Poissonian–like white noises.

An example of independence, conditional expectation and martingales in quantum probability theory is constructed on the permutational Fock space which is a kind of discrete “Fock space” generated by all the permutations from the natural numbers. Besides a discrete time analogue of quantum Ito’s formula for the permutational Fock space is obtained.

It is shown that exponential vectors associated with indicator functions constitute a total set in the boson Fock space Γ (L₂[0,1]). An application to the standard Brownian motion and Poisson process is mentioned.

The interaction between an ultrarelativistic particle and a linear array made up of N twolevel systems (“AgBr” molecules) is studied by making use of a modified version of the Coleman-Hepp Hamiltonian. The energy-exchange processes between the particle and the molecules are properly taken into account, and the evolution of the total system is calculated exactly when the array is initially in the ground state.

We derive an exponential behavior at all times for a certain propagator, in a weak-coupling, macroscopic limit, that is closely related to van Hove’s “λ²T” limit. We subsequently focus our attention on the momentum operator of the particle in the Heisenberg picture and point out the presence of a Wiener process in a restricted portion of the total Hilbert space, in the weak-coupling, macroscopic limit. No assumptions are made on the spectrum of the Hamiltonian of the linear array and no partial trace is computed over its states. We finally discuss the links between the problem of dissipation and the theory of measurement in quantum theory, and consider the possibility of deriving a quantum Langevin equation for our model.

All the results presented are exact.

Using a non-commuting decomposition of the number operator that arises naturally from perturbation of the Poisson process jump times we obtain an extension of the quantum Ito multiplication table and discuss its algebraic representations.

The following sections are included:

• Introduction

• The canonical channel

• Dynamical entropy and couplings

• The sufficiency of the canonical channel

• Sufficiency of the closed pure states channel

• References

We give an algebraic proof for the representation theory of the *– algebra of the quantum group SU_q(2). We find a basis of this *–algebra which is also a basis for q = 0. Then we use the developed notions to give an elementary proof of Woronowiczs result that the C*–algebra $\mathcal{A}$ of SU_q(2) does not depend on q ∈ (−1,1).

Finally, we show that also the domain ${\mathcal{A}}_{\mathcal{T}}\subset \mathcal{A}$ of infinitesimal generators of Lévy processes on SU_q (2) is independent of q ∈ (−1,1). This means that the sets of all infinitesimal generators coincide for all q.

We analyse the thesis of Kümmerer and Maassen (1998) that classical probability is unable to model the the stochastic nature of the Aspect experiment, in which violation of Bell’s inequality was experimentally demonstrated. According to these authors the experiment shows the need to introduce the extension of classical probability known as Quantum Probability. We show that their argument depends on hidden assumptions and a highly restrictive view of the scope of classical probability. A careful probabilistic analysis shows, on the contrary, that it is classical deterministic physical thinking which cannot cope with the Aspect experiment and therefore needs revision. The ulterior aim of the paper is to help mathematical statisticians and probabilists to find their way into the fascinating world of quantum probability (thus: the same aim as that of Kümmerer and Maassen) by dismantling the bamboo curtain between ordinary and quantum probability which over the years has been built up as physicists and pure mathematicians have repeated to one another Feynman’s famous dictum ‘quantum probability is a different kind of probability’.

A misleading ommision in [L] is reinstated below. Definition 2.3 should read as follows…