Karl-Heinz Fichtner

Characterization of states of infinite boson systems. I. On the construction of states of boson systems

In a previous paper [11] it was shown that to each locally normal state of a boson system one can associate a point process that can be interpreted as the position distribution of the state. In the present paper the so-called conditional reduced density matrix of a normal or locally normal state is introduced. The whole state is determined completely by its position distribution and this function. There are given sufficient conditions on a point processQ and a functionk ensuring the existence of a state such thatQ is its position distribution andk its conditional reduced density matrix. Several examples will show that these conditions represent effective and useful criteria to construct locally normal states of boson systems. Especially, we will sketch an approach to equilibrium states of infinite boson systems. Further, we consider a class of operators on the Fock space representing certain combinations of position measurements and local measurements (observables related to bounded areas). The corresponding expectations can be expressed by the position distribution and the conditional reduced density matrix. This class serves as an important tool for the construction of states of (finite and infinite) boson systems. Especially, operators of second quantization, creation and annihilation operators are of this type. So, independently of the applications in the above context this class of operators may be of some interest.

Point processes and the position distribution of infinite boson systems

It is shown that to each locally normal state of a boson system one can associate a point process that can be interpreted as the position distribution of the state. The point process contains all information one can get by position measurements and is determined by the latter. On the other hand, to each so-called Σc-point processQ we relate a locally normal state with position distributionQ.

Quantum Teleportation with Entangled States Given by Beam Splittings

Abstract: Quantum teleportation is rigorously demonstrated with coherent entangled states given by beam splittings. The mathematical scheme of beam splitting has been used to study quantum communication [2] and quantum stochastic [8]. We discuss the teleportation process by means of coherent states in this scheme for the following two cases: (1) Delete the vacuum part from coherent states, whose compensation provides us a perfect teleportation from Alice to Bob. (2) Use fully realistic (physical) coherent states, which gives a non-perfect teleportation but shows that it is exact when the average energy (density) of the coherent vectors goes to infinity. We show that our quantum teleportation scheme with coherent entangled state is more stable than that with the EPR pairs which was previously discussed.

Quantum Teleportation and Beam Splitting

Abstract: Following the previous paper in which quantum teleportation is rigorously discussed with coherent entangled states given by beam splittings, we further discuss two types of models, the perfect teleportation model and non-perfect teleportation model, in a general scheme. Then the difference among several models, i.e., the perfect models and the non-perfect models, is studied. Our teleportation models are constructed by means of coherent states in some Fock space with counting measures, so that our model can be treated in the frame of usual optical communication.

Time Evolution and Invariance of Boson Systems Given by Beam Splittings

Based on a model for general beam splittings we search for states of boson systems which are invariant under the combination of the evolution given by the splitting procedure and some inherent evolution. It turns out that for finite systems only trivial invariant normal states may appear. However, for locally normal states on a related quasilocal algebra representing states of infinite boson systems, one can find examples of nontrivial invariant states. We consider as example a beam splitting combined with a contraction compensating the loss of intensity caused by the splitting process. In general, we observe interesting connections between the splitting procedure and certain thinning operations in classical probability theory. Several applications to physics seem to be natural since these beam splitting models are used to describe measuring procedures on electromagentic fields.

Random permutations of countable sets

SummaryThe main objective of this paper is a study of random decompositions of random point configurations onRd into finite clusters. This is achieved by constructing for each configurationZ a random permutation ofZ with finite cycles; these cycles then form the cluster decomposition ofZ. It is argued that a good candidate for a random permutation ofZ is a Gibbs measure for a certain specification, and conditions are given for the existence and uniqueness of such a Gibbs measure. These conditions are then verified for certain random configurationsZ.

On a Mathematical Model of Brain Activities

The procedure of recognition can be described as follows: There is a set of complex signals stored in the memory. Choosing one of these signals may be interpreted as generating a hypothesis concerning an “expexted view of the world”. Then the brain compares a signal arising from our senses with the signal chosen from the memory leading to a change of the state of both signals. Furthermore, measurements of that procedure like EEG or MEG are based on the fact that recognition of signals causes a certain loss of excited neurons, i.e. the neurons change their state from “excited” to “nonexcited”. For that reason a statistical model of the recognition process should reflect both—the change of the signals and the loss of excited neurons. A first attempt to explain the process of recognition in terms of quantum statistics was given in [1]. In the present note it is not possible to present this approach in detail. In lieu we will sketch roughly a few of the basic ideas and structures of the proposed model of the ...

Asymptotic behaviour of time evolutions of infinite particle systems

SummaryMotions of one-dimensional infinite particle systems are considered where the dynamics is given by systems of ordinary differential equations of first order. The aim of the paper is to show that under certain assumptions about the system of differential equations the distribution law Ptof the particle system at time t becomes more and more regular under the influence of such an interaction. Moreover, Ptis tending weakly toward a distribution describing a random particle system with equal successive spacings.

On a Quantum‐Like Model of the Recognition Process

We deal with a quantum‐like model of the recognition process. The model was developed in the past years in a series of papers (cf. [1, 3, 4, 5, 6]). The present paper which is a continuation of [7] is focussed on a relation between recognition and a certain process of self‐collapses. It is shown that the process of self‐collapses can be described by a classical homogenous discrete‐time Markov chain. Finally we pass over to a continuous‐time Markov chain that seems to give a more appropriate and reasonable characterisation of the process of recognition.

Teleportation schemes in infinite dimensional Hilbert spaces

The success of quantum mechanics is due to the discovery that nature is described in infinite dimension Hilbert spaces, so that it is desirable to demonstrate the quantum teleportation process in a certain infinite dimensional Hilbert space. We describe the teleportation process in an infinite dimensional Hilbert space by giving simple examples.