Quantum Probability and Infinite Dimensional Analysis

Cottbus, 15 - 20 March, 2001

Amosov, Grigori Moscow Institute of Physics and Technology

Toward a Wold decomposition for stationary quantum stochastic processes

Consider a C_{0}-semigroup of nonunitary isometries V = (V_{t})_{t∈R_{+}} in a Hilbert space H. Then the Wold decomposition H = H_{1} ⊕ H_{2} defines the maximum unitary part V_{1} = V |_{H_{1}} and the maximum completely nonunitary (shift) part V_{2} = V |_{H_{2}} of the semigroup V. Analogously it can be done a Wold decomposition of the stationary stochastic process X_{t} = X_{t}^{(1)} ⊕ X_{t}^{(2)}, t ∈ R, on the deterministic part X^{(1)} = (X_{t}^{(1)})_{t∈R} and the completely nondeterministic part X^{(2)} = (X_{t}^{(2)})_{t∈R}. Let ((j_{t})_{t∈R}, M) be a pair consisting of homomorphisms j_{t} : L^{∞} → M, t ∈ R, where M is an algebra of all functionals from the process X, defined by the formula j_{t}(f) = f(X_{t}), f ∈ L^{∞}, t ∈ R. Then M = M_{1} ⊗ M_{2}, where M_{i} is an algebra of all functionals over X^{(i)}, i = 1, 2, can be called a Wold decomposition for the process X on the algebraic level. Due to L. Accardi, A. Frigerio and J.T. Lewis one can look at the endomorphic semigroup α = (α_{t})_{t∈R_{+}} on the von Neumann algebra M as a stationary quantum stochastic process. We point out that if the cocycle conjugacy class of α containes the flow of shifts, one can define a Wold decomposition of M associated with α which is an analogue of the Wold decomposition for classical stochastic processes on the algebraic level. This decomposition defines two von Neumann subalgebras M_{1}, M_{2} ⊂ M generating M such that α|_{M_{1}} consists of automorphisms and α|_{M_{2}} is a flow of shifts. All results illustrated for the quasifree endomorphic semigroups on the CAR algebra using the technics of [1-2].

References:

[1] G.G. Amosov, J. Russ. Math., 2000, No 2, P. 7-12.

[2] G.G. Amosov, Infinite dimensional analysis, Quantum Probability and Rel. Top.

- 2000. - V.3, No 2. - P. 237 -246.

Bhat, B. V. Rajarama Indian Statistical Institute, Bangalore, INDIA.

E_{0}-semigroups and dilation theory

We study the partially ordered set of quantum dynamical semigroups dominated by a given semigroup on the algebra of all bounded operators on a Hilbert space. For E_{0}-semigroups this set can be described through cocycles. This helps us to prove a factorization theorem for dilations. These cocycles can be completely parametrized in the type I case, and this allows us to settle some minimality questions. Most of the results can be generalized to general C^{∗}-algebras by considering product systems of Hilbert modules.

Szynal, Dominik and Bieniek, Mariusz Institute of Mathematics, UMCS Lublin, Poland

Characterizations of distributions via linearity of regression of non–adjacent order statistics

Let X(r, n, m̃, k), r = 1, 2, ..., n, denote generalized order statistics based on an absolutely continuous distribution function F (cf. [1]). We characterize those F for which the regression E (X(r + i, n, m̃, k) | X(r, n, m̃, k) = x) or E (X(r, n, m̃, k) | X(r + i, n, m̃, k) = x) where i ≥ 1, r + i ≤ n, is a linear function of x.

References:

[1] Kamps, U., A concept of generalized order statistics, B. G. Teubner, Stuttgart, 1995.

Cecchini, Carlo Dipartimento di matematica ed Informatica, Udine University

Canonical maps on von Neumann algebras

A unified treatment of generalized conditional expectations, Radon Nikodym theorems and operator valued weights on von Neumann algebras is presented. Namely, given two von Neumann algebras on which two normal faithful semifinite weights are defined, a canonical map between the two algebras which intertwins the two weights is defined, of which the above classical mappings are particular cases.

Chebotarev, A. M. 119899 Moscow, MSU, Quantum Statistics Dept.

Sufficient Selfadjointness Criteria for Symmetric Boundary Value Problems Related to QSDE

A generic quantum measurement is implemented as the measuring of parameters of the current of massless (photons) or massive particles scattering on a localized quantum system. The case, when the scale of the quantum system is incomparably small with respect to the scale of the current, the interaction between the current and quantum system can be considered as singular. In some important cases, this interaction can be described by a symmetric boundary value problem in Fock which is unitary equivalent to QSDE in the Hudson–Parthasarathy sense or its analogous with new types of nonadapted fundamental quantum noises. We suggest conditions which are sufficient for some classes of boundary-value problems to be well-posed selfadjoint problems with trivial deficiency index.

Emel’yanov, Eduard Mathematisches Institut der Universität Tübingen

Some asymptotical properties of positive operators on noncommutative L_{1}-spaces

The presentation of some recent results obtained by author jointly with Manfred Wolff (University of Tübingen) are given. They generalize previous results of A. Lasota, J. Komornik etc. on Markov operators on commutative L^{1}-spaces. Let T = (T_{t}) be a discrete or strongly continuous semigroup of positive operators on the predual M_{∗} of a W^{∗}-algebra M. A ⊆ (M_{∗})_{sa} is called a real attractor for T if

lim_{t→∞} dist(T_{t}x, A) = 0 for each x ∈ (M_{∗})_{sa}, ∥x∥ ≤ 1. (M_{∗})_{+} ∋ h ≠ 0 is called a nontrivial

lower-bound element for T whenever lim_{t→∞} ∥(h − T_{t}f)^{+}∥ = 0 for all 0 ≤ f ∈ M_{∗}, ∥f∥ = 1.

Among others we present here the following results:

i) (Attractor correction theorem) Let [−y, y] + ηB_{(M_{∗})_{sa}} be a real attractor for T and 0 ≤ η < 1. There exists a positive T -fixed w ∈ M_{∗}, ∥w∥ = ∥y∥ such that (1 − η)^{−1}[−w, w] is a real attractor for T. ii) Let T consist of Markov operators. There exists a (positive T -fixed) u ∈ M_{∗} such that lim_{t→∞} T_{t}f = f(1)u for each f ∈ M_{∗} iff T has a nontrivial lower-bound element.

iii) Let T consist of completely positive contractions and possesses a quasi order bounded real attractor. Then T is weakly constrictive. Moreover, if M is atomic or of the form L^{∞}(X, Σ, μ) ⊗ N where N is a finite dimensional C^{∗}–algebra then T is constrictive.

Fagnola, Franco University of Genova

Quantum Markov semigroups and their stationary states

We discuss some tools for studying the qualitative behaviour of a class of Quantum Markov Semigroups (QMS) on B(h). These are w^{∗}-continuous semigroups of completely positive normal maps with infinitesimal generator associated with a quadratic form

L̃(x)[v, u] = ⟨Gv, xu⟩ + ∑_{ℓ=1}^{∞}⟨L_{ℓ}v, xL_{ℓ}u⟩ + ⟨v, xGu⟩

where G is the infinitesimal generator of a strongly continuous contraction semigroup on h, L_{ℓ} are operators on h with dom(L_{ℓ}) ⊇ dom(G), v, u ∈ dom(G) and L̃(1)[v, u] = 0, (1 is the identity operator on h). These semigroups appear in the study of irreversible evolutions of quantum open systems (see e.g. [1], [4]). In this talk we present some recent results on the existence, uniqueness and faithfullness of invariant states leading to a method for studying the qualitative behaviour of the above evolutions as a non-commutative dynamical system. We apply this method to the study of some evolutions (master equations) arising in Quantum Optics (see e.g. [4]). These are illustrated in the physical literature only via numerical simulations.

References

[1] Alicki, R.; Lendi K. Quantum Dynamical Semigroups and Applications. Lecture Notes

in Physics, 286. Springer (1987).

[2] Fagnola, F.; Rebolledo, R. On the existence of stationary states for quantum dynamical semigroups. To appear in J. Math. Phys. 2001.

[3] Fagnola, F.; Rebolledo R.: Subharmonic projections for a quantum Markov semigroup. Preprint. January 2001.

[4] Gisin, N.; Percival, Ian C. The quantum-state diffusion model applied to open systems. J. Phys. A 25 (1992), no. 21, 5677–5691

Franz, Uwe EMAU Greifswald

Braided independence and braided Lévy processes

We define and study Lévy processes and Brownian motion on braided spaces. First, we introduce the necessary notions and generalize some fundamental results, in particular about the positivity of the convolution of positive functionals and the one-to-one correspondences between generators, convolution semi-groups and Lévy processes. Next we exhibit a construction of examples of braided ∗-Hopf algebras, and also give their symmetrization. Finally, we study Brownian motion, i.e. Lévy processes with quadratic generators, on these algebras. We show how Brownian motions on these braided ∗-spaces can be classified and we give the quantum stochastic differential equations that define a realisation of these processes on boson Fock space.

Freudenberg, Wolfgang Brandenburgische Technische Universität Cottbus

On quantum logical gates on the Fock space

The so-called FTM gate is composed of two input gates I_{1}, I_{2}, and one control gate C. Two inputs come to the first splitting apparatus and produce two outputs. One of them passes via a control gate to the second beam splitter, the second output (of the first beam splitting procedure) passes directly to the second beam splitter. After the second beam splitting we will have two final outputs O_{1}. O_{2}. We will give examples for quantum channels transmitting some information such that the gate will fulfill the usual truth table. Hereby, the signals 0 and 1 will be encoded by several classes of states. The control gate if switched on will change the information (0, 1) into (1, 0) but leaves the information (0, 0) resp. (1, 1) unchanged. We will consider quantum channels for states on the symmetric Fock space over a general metric space G. Instead of the usual beam splitting with constant splitting rates we will use more general splitting procedures.

Gohm, Rolf EMAU Greifswald

Prediction Errors and Completely Positive Maps

Prediction for a stationary stochastic process is considered. We give an operatortheoretic approach by introducing the concept of an adapted isometry. This is an abstract characterization of the time evolution of a stationary process adapted to a well-behaved filtration. A product decomposition for adapted isometries then opens up the possibility of a step-by-step analysis. We define a sequence of associated completely positive maps in such a way that the non-linear, one-step, mean-square prediction errors of the process with respect to the filtration can be expressed by products of these maps. In other words, the evolution of the prediction errors in time can be described by a sort of dynamics which is well-known in quantum theory as a irreversible dynamics of mixed states. Looking in particular at large time periods we find that determinism for prediction corresponds to an absorbing state for this dynamics.

Goswami, Debashish Alexander von Humboldt Fellow, Inst. Für Angewandte Math., Univ. of Bonn

Evans-Hudson Dilation for a class of quantum dynamical semigroups with unbounded generator

The central problem addressed in the talk is to describe a canonical construction of Evans-Hudson (E-H) flow which dilates a given quantum dynamical semigroup. Let A be a C^{∗} or von Neumann algebra acting on a Hilbert space h and (T_{t})_{t≥0} be a quantum dynamical emigroup on A, i.e. a C_{0}-semigroup of completely positive and contractive (and normal if A is von Neumann algebra) maps acting on A. We say that a family (j_{t})_{t≥0} of ∗-homomorphisms from A into A ⊗ B(Γ(L^{2}(R_{+}) ⊗ k_{0})) for some Hilbert space k_{0} (where Γ stands for the symmetric Fock space) is an E-H dilation of T_{t} if T_{t}(x) is obtained from j_{t}(x) by taking the vacuum conditional expectation w.r.t. the above Fock space, and furthermore j_{t} satisfies an E-H type quantum stochastic flow equation on some dense ∗-subalgebra(say A_{∞}) of A (cf. [KRP] for the definition of an E-H flow). In this present talk we shall describe various existence results for obtaining an E-H flow for a given quantum dynamical semigroup. First of all, we mention that such a dilation is always possible if the underlying algebra is either a von Neumann algebra or a separable unital C^{∗}-algebra and the generator of the given semigroup is norm-continuous (cf [GS], [GSP]). However, the norm- continuity is a rather strong assumption and is not valid in many interesting situations. To relax this restriction, we consider an additional structure in our set-up, namely the action of a Lie group G on the algebra such that the given semigroup is covariant under this action. In this case, we shall show that if one makes some more reasonable assumptions regarding the generator of the semigroup (but not too strong like normcontinuity) then it is possible to ensure the existence of E-H dilation, which will also be covariant with respect to an appropriate G-action on the relevant space. We also discuss the possibility of implementing this EH flow by unitary or partial-isometetry valued operator processes obtained by solving a suitable Hudson-Parthasarathy type q.s.d.e. with unbounded operator coefficients. Applications of these results include the semigroups associated with interesting classical stochastic processes,in particular the heat semigroup. If time permits, we also focus on the relation of this theory with the Arveson-Powers index theory for quantum dynamical semigroups.

References:

1. [GS] D. Goswami and K. B. Sinha; Hilbert Modules and Stochastic Dilation of a Quantum Dynamical Semigroup on a von Neumann Algebra, Comm. Math. Phys., vol 205 (no. 2), 377-403 (1999).

2. [GSP] D. Goswami, K. B. Sinha and A. Pal; Existence of Evans-Hudson Dilation in a C^{∗}-algebraic Set-up, Inf. Dim. Anal. Quan. Prob. and Rel. Topics, World Scientific, vol. 3 (no. 1), 177-184(2000).

3. [KRP] K.R. Parthasarathy, ”An introduction to quantum stochastic calculus”, Monographs in Mathematics, 85. Birkhäuser Verlag, Basel, 1992.

4. D. Goswami and K. B. Sinha; Dilation of semigroups with unbounded genetaror, in preparation.

Gregoratti, Matteo Dipartimento di Matematica del Politecnico di Milano

The Hamiltonian Operator associated to some Quantum Stochastic Differential Equations

We consider the Hamiltonian operator associated to the quantum stochastic differential equation introduced by Hudson and Parthasarathy to describe a quantum mechanical evolution in presence of a “quantum noise”. We characterize such Hamiltonian in the case of arbitrary multiplicity and bounded coefficients: we find an essentially self-adjoint restriction of the operator and, in particular, we provide an explicit construction of vectors belonging to its domain. This restriction of the Hamiltonian is always an operator of the Chebotarev’s kind.

Madalin, Guta University of Nijmegen

Generalised Brownian motion

Abstract A generalised Brownian motion associates in a functorial way to any (separable) Hilbert space an algebra of creation and annihilation operators represented on a Fock-like Hilbert space with a prescription for the calcuation of moments for the vacuum state in terms of a function on pair partitions. The positivity requirement is encoded by the ^{∗}−semigroup of “broken pair partitions”. We define a q−product of such Brownian motions which interpolates between the tensor and free product, and describe the corresponding central limit theorem.

Hermisson, Ulrich University Greifswald

Differential calculus on quantum spaces.

The notion “quantum space” and the notion “covariant differential calculus” introduced by S. L. Woronowicz are reviewed. A general construction of covariant differential calculi is given which exhibits quantum partial derivatives. The 2-dimensional covariant differential calculi on the quantum 2-sphere are classified by use of a general method.

Hudson, Robin L. Department of Mathematics, Nottingham Trent University

Universal solutions of the quantum Yang-Baxter equation constructed as symmetrised double quantum stochastic integrals

We consider a Lie algebra denoted by dL which is got by taking commutators in an associative algebra. One may think of dL as labelling the basic integrators of quantum stochastic calculus, but in fact what follows is independent of such calculus. We show that there is an intrinsic differential calculus on the universal enveloping algebra U over dL. Using this we construct double product integrals G(h) = ∏ ∏(1 + dg(h)), where dg(h) is a formal power series in the indeterminate h with vanishing constant term and coefficients in the exterior product of two copies of L, which are formal power series with coefficients in the tensor product of two copies of U. It is conjectured that a necessary and sufficient condition that G(h) satisfy the quantum Yang-Baxter equation

G(h)^{12}G(h)^{13}G(h)^{23} = G(h)^{23}G(h)^{13}G(h)^{12}

is that dg(h) satisfy

[dg(h)^{12}, dg(h)^{13}] + [dg(h)^{12}, dg(h)^{23}] + [dg(h)^{13}, dg(h)^{23}]
+dg(h)^{12}dg(h)^{13}dg(h)^{23} − dg(h)^{23}dg(h)^{13}dg(h)^{12}
= 0.

This condition implies that the lowest order coefficient dg_{1}of dg(h) satisfies the classical Yang-Baxter equation, and, given such dg_{1}, gives a hierarchy of finite dimensional linear equations for the higher order coefficients. It is conjectured that this hierarchy always has solutions. An abstraction to this context of the theory of quantum stochastic flows gives another method for constructing deformations of universal enveloping algebras.

Inoue, Kei Science University of Tokyo, Japan

A description of quantum chaos by chaos degree

There exist several approaches in the study of chaotic behavior of dynamical systems. But these concepts are rather independently used in each field. In 1991, M.Ohya proposed Information Dynamics (ID for short) to treat such chaotic behavior of systems altogether. A chaos degree to measure the chaos in classical dynamical systems is defined by means of a complexity in ID. In particular, among several chaos degrees, the entropic chaos degree is applied to some dynamical systems. In this talk, we study the chaotic behaviour and the quantum-classical correspondence for the baker’s map. The chaos degree is computed and it is demonstrated that it describes the chaotic features of the model. The correspondence between classical and quantum chaos degrees is considered. This study is joint work with Prof.Ohya and Prof.I.V.Volovich.

Kläre, Steffen GSF Forschungszentrum für Umwelt und Gesundheit GmbH

On a class of linear stochastic differential equations without adaptedness

We consider certain stochastic differential equations and the corresponding equations in terms of boson Fock space. Using some elementary instruments from point process theory we solve these equations. Examples generalizing the geometric Brownian motion will be discussed.

Kozyrev, Sergei Centro Vito Volterra & Semenov Institute of Chemical Physics

Quantum Boltzmann statistics in interacting systems

Collective operators for generic quantum system 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.

Kümmerer, Burkhard Mathematisches Institut A, Universität Stuttgart

Markov Proesses and the Preparation of Quantum States

A Micro-Maser can be viewed as a realization of a quantum Markov-process in its coupling representation: The field mode, as an quantum system, is coupled to an infinite sequence of incoming atoms. Asymptotic completeness of such a Markov process can be reformulated into the property that the incoming atoms can generated an arbitrary state of the open quantum systems. A theoretical discussion of such systems is followed by some numerical results for the Micro-Maser, which demonstrate its physical significance.

Lenczewski, Romuald Wroclaw University Of Technology

A stochastic integral representation of certain noncommutative random variables

Following the GNS construction for the hierarchy of freeness, we will show that various noncommutative random variables can be represented as stochastic integrals with respect to projection-valued measures for various notions of noncommutative independence. Spectral measures will be given for free, m-free and similar to free random variables.

Lindsay, J. Martin School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD

Markovian cocycles on C^{∗}-algebras

This talk is concerned with the functional equation

k_{s+t} = k̂_{s} ∘ σ_{s} ∘ k_{t}

for quantum stochastic evolutions on a C^{∗}-algebra, and infinitessimal descriptions through quantum stochastic differential equations

dk_{t} = k_{t} ∘ ϕ_{β}^{α} dΛ_{α}^{β}(t).

We shall take a brief tour of some very pleasant ‘quantum functional analysis’ which is proving invaluable in deepening our understanding of both equations. In particular, this analysis allows a fruitful extension of these equations to the context of operator spaces and operator systems, and this will be the point of departure for Stephen Wills’ lecture.

References:

[1] Existence of Feller cocyles on a C^{∗}-algebra (with S J Wills), Bull London Math Soc (to appear).

[2] Existence, positivity and contractivity for quantum stochastic flows with infinite dimensional noise (with S J Wills), Probab Th Related Fields 116 (2000), 505–543.

[3] On the generators of quantum stochastic flows (with K R Parthasarathy), J Funct Anal 158 (1998), 521–549.

[4] Markovian cocycles on operator algebras, adapted to a Fock filtration (with S J Wills), J Funct Anal 178 (2000), 269–305.

[5] A stochastic Stinespring theorem (with D Goswami and S J Wills, Math Ann (2000).

[6] Dilations of Markovian cocycles on a von Neumann algebra (with D Goswami, K B Sinha and S J Wills), Pacific J Math (submitted).

[7] Markovian cocycles on operator spaces (with S J Wills), in preparation.

Nou, Alexandre Laboratoire de Mathematiques, Universite de Franche-Comte

Non injectivity of the von Neuman algebra generated by q-Gaussians

In this talk we discuss the non-injectivity of the von Neumann algebra generated by q-gaussians acting on the q-deformed Fock space F_{q}(H), where q ∈] − 1, 1[ and H is some real Hilbert space. Our main result is a dimensionless extension of a result of M.Bozejko and R.Speicher in [1]. To establish it we use a generalization of a well known inequality due to U.Haagerup and M.Bozejko [2], which is now valid in the q-deformed case and for operator coefficients.

References

[1] M.Bozejko, R.Speicher, Completely positive maps on Coxeter Groups, deformed commutations relations, and operator spaces, Math. Annalen 300 (1994), 97-120.

[2] M. Bozejko, Ultracontractivity and Strong Sobolev Inequality for q-Ornstein-Uhlenbeck semi-

group (-1¡q¡1), Infinite Dim. Analysis, Vol.2, No 2 (1999) 203-220.

Obata, Nobuaki Graduate School of Mathematics, Nagoya University

Quantum decomposition and applications to asymptotic spectral analysis on large graphs

Motivated by the work of Accardi–Bożejko [1] on relation between orthogonal polynomials and interacting Fock spaces, we introduce “quantum decomposition” of a classical random variable. With this idea asymptotic spectral distributions of adjacency matrices (Laplacians) of large graphs are determined directly as a consequence of quantum central limit theorems. Preliminary results have been reported in [2,3] and further study is now in progress with Y. Hashimoto and A. Hora.

References

[1] L. Accardi and M. Bożejko: Interacting Fock spaces and Gaussianization of probability measures, Infinite Dimen. Anal. Quantum Prob. 1 (1998), 663–670.

[2] Y. Hashimoto: Quantum decomposition in discrete groups and interacting Fock spaces, preprint, 2001.

[3] Y. Hashimoto, N. Obata and N. Tabei: A quantum aspect of asymptotic spectral analysis of large Hamming graphs, in “Quantum Information III (T. Hida and K. Saitô, Eds.),” World Scientific, 2001 (in press).

Oravecz, Ferenc Alfred Renyi Mathematical Institute, The Hungarian Academy of Sciences Realtanoda

”Poissonization”

We define a natural transformation of certain symmetric probability measures, and show that it brings the Gaussian-law as well as several deformed analogue of it to the Poisson-law or to the corresponding deformed Poisson-law, respectively. An important example is the q-Gaussian-law with its transform being the q-Poisson-law of Saitoh and Yoshida. By analogy we define an even more general class of deformed Poisson distributions, and show that, as in the classical and the q-case, these deformed Poisson-laws can be achieved as the distributions of certain operators.

Privault, Nicolas Universite de La Rochelle and Universidade de Lisboa

Renormalized splitting of Poisson noise and Lévy processes on Lie algebras

The compensated Poisson noise is expressed as a composite sum (splitting) of creation and annihilation operators, whose probabilistic interpretation relies on time changes. We construct a renormalized Itô table for this decomposition and obtain some continuous time realizations of Lévy processes on the finite difference algebra and on sl_{2}, e.g. the space-time dual of the Poisson process (or compensated gamma process), and the continuous binomial process.

Schubert, Katrin Brandenburg Technical University of Cottbus

A Class of Quantum Markov Chains on the Fock Space

Using two operators closely related to the Malliavin derivative and the Skorohod integral we define a transition expectation that relates to each initial normal state on the Fock space a homogenous quantum Markov chain. We characterize the chain of position distributions related to such a quantum Markov chain in general and illustrate this with a special class of examples.

Skeide, Michael BTU Cottbus

Central Units, the Christensen-Evans Theorem, and Type I Product Systems of Hilbert Modules

1. Positive definite kernels

2. Completely positive definite kernels

3. Product systems

4. Time ordered Fock module

5. Central units 6. Index of spatial product systems

A. M. Chebotarev and A. V. Tchourkin, 119899 Moscow, MSU, Quantum Statistics Dept.

On Resonant Standard Quantum Limits For Gravitational-Wave Antenna

In quantum measurements theory, the standard quantum limits (QSL) characterize the bounds for the sensitivity of a realistic measuring device. For a solvable model of gravitational-wave antenna, we find the estimates of standard quantum limits for detectable amplitude of the perturbations of the metric tensor induced by the incident periodic gravitational wave in the resonant and nonresonant cases. The perturbation of the metric is measured by the laser interferometer. Classical and exponential distributions of number of photons of the laser light are considered. We point out that in the case of exponentially distributed intensity of the coherent light, the estimates of SQL strongly depend on the choice of an observable. Numerical values of QSLs are discussed for the LIGO device.

Uhlmann, Armin Institut f. Theoretische Physik, Univ. Leipzig

Anti-linearity in bi-partite quantum systems

Abstract The direct product H_{1} ⊗ H_{2} of two Hilbert spaces is naturally isomorphic to the anti-linear (Hilbert Schmidt) maps from H_{1} into H_{2}. This well known fact is quite directly related to the Einstein, Podolski, Rosen effect. It enables a neat description of quantum teleportation and of similar issues. There is an elementary (as we are concerned with type I) link to the modular conjugations Wilde, Ivan Department of Mathematics, King’s College London

Quantum stochastic integrals in the plane.

We construct stochastic integrals in the plane using quantum martingale integrators. This provides an extension of the theory of Wong and Zakai to a quantum context.

Wills, Stephen University of Nottingham

Characterising CP contraction cocycles

We report on extensions of the recent characterisation by Accardi and Kozyrev of CP unital cocycles on B(H) with one dimension of noise to the case of CP contraction cocycles on either an operator system or a nonunital C^{∗}-algebra, and arbitrary noise dimension space.