Stochastics and Quantum Physics

Workshop on Stochastics and Quantum Physics October 21-26, 1999 Centre for Mathematical Physics and Stochastics MaPhySto University of Aarhus 1 Introduction The Workshop focused on some of the areas where concepts and techniques from Stochastics (i.e. Probability and Mathematical Statistics) are, or seem likely soon to become, of real quantum physical importance. By bringing together leading physicists and mathematicians, having an active interest in the themes of the Workshop, it was sought to foster fruitful discussions and collaboration on the role and use of Stochastics in Quantum Physics. In this leaflet we have gathered the (extended) abstracts of the talks given. We thank all contributors for taking upon them the extra work of writing these extended abstracts. We hope that this booklet may be of some use to mathematicians as well as physicists working in the area of interplay between Quantum Physics and Stochastics. At the end of the booklet, the schedule of the workshop and the list of participants is reproduced. We wish to thank all participants the speakers in particular for contributing to the workshop. Ole E. Barndorff-Nielsen and Klaus Mølmer. 1

Contents 1 Introduction 1 2 (Extended) Abstracts of Talks 3 LuigiAccardi... 3 AlbertoBarchielli... 8 FrançoisBardou... 17 ViacheslavP.Belavkin... 22 HowardCarmichael... 22 AlexanderGottlieb... 23 IngeS.Helland... 33 AlexanderS.Holevo... 37 PeterHøyer... 48 Uffe Haagerup.... 49 Göran Lindblad... 49 YuriYu.Lobanov... 50 ElenaR.Loubenets... 56 HansMaassen... 62 GüntherMahler... 72 SergeMassar... 72 IanPercival... 73 AsherPeres... 78 DénesPetz... 84 HowardWiseman... 89 JeanClaudeZambrini... 94 BerntØksendal... 99 3 Workshop Program (revised) 104 4 List of participants 108 2

2 (Extended) Abstracts of Talks The abstracts/papers are ordered alphabetically after the last name of the author who presented the work. Luigi Accardi The stochastic limit of quantum theory and the dilation problem. My task was to discuss the connections between the stochastic limit of quantum theory and the dilation problem. The dilation problem is the following: given a Markov semigroup to construct a Markov process whose canonically associated semi group is the given one. The stochastic limit studies the following problem: given a Hamiltonian system (classical or quantum), depending on a parameter λ, study the behaviour of this system in a time scale of order t/λ 2. In the case of interest, the parameter λ is small and therefore the stochastic limit is related to long time scales (as scattering theory). On the other hand the smallness of the parameter λ reflects a weak interaction (as in perturbation theory). Thus the stochastic limit is a new asymptotic technique in the study of dynamical (Hamiltonian for the moment) systems, combining together scattering and perturbation theory with the new ideas on quantum Markov processes, stochastic calculus, central limit theorems, which arose from quantum probability. This mixture brought a multiplicity of results both in physics [AcLuVo00] and in mathematics (cf. [AcLuVo97] for the notion of interacting Fock space and Fock module, [Ske99] for their relationships, [AcLuVo99] for the white noise approach to classical and quantum stochastic calculus). Apparently the two topics are far apart, but there is a connection: the unique feature of the stochastic limit with respect to all the up to now known classical and quantum asymptotic methods, is that: the dominating contribution (in a suitable topology) to the unitary evolution is still a unitary evolution. Even more: it is a unitary Markovian cocycle and therefore, by a (now standard) technique introduced in [Ac78] it allows to construct a Markov process and a Markov semigroup. Thus, using the language of dilations we could say that in the stochastic limit, the original Hamiltonian evolution converges to a unitary dilation of a Markov semi group. A first natural question is: which Markov semigroups can be obtained with the stochastic limit technique? The answer is: all those semigroups of which a dilation can be constructed by means of classical or quantum stochastic calculus. One would like to have a The contribution of Richard D. Gill has appeared in the separate note Asymptotics in Quantum Statistics, Miscellanea No. 15, October 1999, Centre for Mathematical Physics and Stochastics, University of Aarhus. Unfortunately L. Accardi had to cancel his participation in the workshop. This is the manuscript for the talk he would have given. 3

definitive result of the type: all the Markov semigroups admit a dilation obtained through the stochastic limit. Up to now the obstruction to such a result was that an infinitesimal characterization of isometric flows, analogue to the Stone theorem for strongly continuous unitary groups or to the Hille Yoshida theorem for C 0 semigroups, was absent. Recently such a characterization has been obtained in [AcKo99b] where it is proved that, if B is an arbitrary C algebra, any completely positive flow on the space B Γ(L 2 (R)) (Γ 2 (L 2 (R)being the Boson Fock space on L 2 (R)) is characterized by a single, completely positive (but non Markovian), semigroup on M 2 (B), the 2 2 matrices with coefficients in B. This extended semigroup is strongly continuous if and only if the associated flow has this property. Therefore this theorem reduces the classification of strongly continuous completely positive flows to the known classification of C 0 semigroups, achieved via the Hille Yoshida theorem. This result, combined with the stochastic golden rule, allowed to give what seems to be the first deduction of the flows associated to the Glauber Kawasaki type dynamics, and in fact of all the dynamics used in the theory of interacting particle systems, from a Hamiltonian model, as well as a single unified proof of the existence of such flows in arbitrary dimensions [AcKo99a]. This result is new even when restricted to classical (commutative) flows which, as it is well known, include all the classical stochastic processes which satisfy a stochastic differential equation driven by Wiener or compound Poisson processes. Another natural question is: from the point of view of physics, what is the relation between a dilation constructed by stochastic calculus and one obtained by the stochastic limit? The answer is simple: the same relation existing between a phenomenological model and a physical law, deduced by basic principles. In fact, given a Markov semigroup, one can aprioriinvent uncountably many unitary dilations of it: classical, Boson, Fermi, free, q deformed, Fock, finite temperature, squeezing,... Moreover, even if we want to restrict ourselves to the Boson Fock case, still there are uncountably many choices which can be made and which are completely equivalent from a purely mathematical point of view. The usual constructions, in the physical literature, of dilations of Markov semigroups are based on the following steps: i) one starts from the generator of a Markov semigroup (master equation) ii) on the basis of more or less plausible physical arguments, one chooses, among the infinitely many unitary dilations of it, a definite one iii) one describes some physical phenomena using this dilation. This procedure is not very satisfactory from the physical point of view for the following reasons: i1) The master equation is itself an approximation, so it should be one of the final results of the construction of a model and not its starting point. i2) The structure of the noise itself, driving the stochastic equation, used to construct the dilation, has a deep physical meaning which cannot be invented, but is one of the essential characteristics to be deduced from the physical model. 4

The stochastic limit bypasses these problems because it starts from the well established Hamiltonian models of quantum physics and to each of them it associates in a unique way a unitary (or isometric) flow and to this, by the quantum Feynman Kac formula of [Ac78], a Markov semigroup. The flow is the limit of the Heisenberg evolution (in interaction representation) of the original Hamiltonian system and the Markov semigroup is the limit of the expectation of the flow for the reference state of the fast degrees of freedom of the system (reservoir, environment, field, gas,...). This means that the Heisenberg equation of motion converges to a (stochastic) Langevin equation whose expectation gives the master equation. It follows that all the parameters which enter in these equations have a microscopic interpretation and can be experimentally controlled. In the class of physical systems to which the stochastic limit technique can be applied, one can distinguish 3 levels in increasing order of difficulty: Level I corresponds to the standard open system scheme, i.e. a discrete spectrum system interacting with a continuous spectrum one. It should be noted that all the models studied up to now in the physical literature concern this level. The situation in the remaining two levels is too complex to be handled by plausibility arguments. For the models in this class the stochastic golden rule gives to the physicists a simple recepee which allows to solve, with very few elementary calculations, the following problem: given the Hamiltonian model, how to write the stochastic Schrödinger equation obtained by taking the stochastic limit? Since this rule is extremely simple to apply and since all the master equations which are the (more or less implicit) starting point of the unitary dilations built up to now in the physical literature, presuppose an underlying Hamiltonian model, the stochastic limit procedure offers to the physicist the opportunity to replace the, up to now standard, scheme: Hamiltonian system master equation dilation by the stochastic limit scheme: Hamiltonian system dilation master equation which is much more satisfactory because now not only the master equation, but also the noise and the Langevin equation become uniquely determined by the original Hamiltonian system. Level II has to do with the low density limit and is much more difficult than Level I because, while in case of Level I the stochastic golden rule gives the possibility to guess the correct stochastic equation by simple inspection of the first and second order terms of the iterated series (which, in the limit, give respectively the martingale and the drift term of the stochastic equation), in the case of Level II, the drift term of the stochastic equation receives contributions from each term of the iterated series and to single out these contributions and resum them into the 2 particle scattering operator is a subtle point. Level III replaces the discrete continuum interactions of the first two levels by continuum continuum ones. Here dramatically new phenomena arise, the most important of which is 5

the breaking of the commutation (or anticommutation) relations and the subsequent replacement of the Fock space by the interacting Fock space and emergence of Fock modules. Another non trivial point is the emergence of new statistics based on non crossing diagrams rather than on the usual boson or fermion ones (in which all crossing diagrams are allowed). The mathematical interpretation of these diagrams in terms of free independence as well as their connection with the semi circle law was discovered by Voiculescu. The stochastic limit gave rise to more sophisticated and more physically interesting notions of independence in which the role of the Gaussian is played by different measures whose explicit form is, in many cases, still unknown (even if all their momenta) can be written explicitly. The absence of explicit formulae is one of the common features of nonlinear problems: the semicircle law corresponds to a linear problem (absence of interaction, in physical terms) and, as usual in linear problems, in this case all calculations can be made explicitly. The fact that the non crossing diagrams give the dominating contribution to the quantum dynamics was first discovered, in the case of QED without dipole approximation, in the paper [AcLu92] and the fact that in the huge literature devoted to this topic (surely much larger in volume than that devoted to the last Fermat problem and involving people such as Dirac, Fermi, Heisenberg, Landau,...) such an important phenomenon was not even conjectured, is an indication of how hidden it was. In fact the original proof was rather elaborated but a much simpler and intuitive one was given later in [AcKoVo98] where the following intuitive picture was derived: before the stochastic limit (finite coupling constant λ) by effect of the nonlinearity the time rescaled fields obey a q commutation relation with the constant q depending both on time and on λ. In the stochastic limit (λ 0) this quantity tends to zero (this give an intuitive explanation of why only the non crossing diagrams survive). The explanation of the Hilbert module structure and the explicit form of the (new type of) quantum stochastic equation requires more work and a good reference for this is Skeide s paper [Ske99]. From the paper [AcLu92] several new mathematical notions emerged: the notion of full Fock module (in a particular case: the general case was dealt with one year later by Pimsner), the notion of interacting Fock space, of stochastic integration on Hilbert module (mathematically developed by Lu and later by Speicher and Skeide), the white noise approach to stochastic calculus on Boltzmannian interacting Fock space (which includes the free case). In particular the notion of interacting Fock space turned out to have a multiplicity of unexpected connections with apparently totally unrelated fields of mathematics such as orthogonal polymomials, wavelets, solvable models in statistical mechanics, new forms of independence and of central limit theorems,... Among the new physical implications of this paper we mention the power decay law in the polaron model [AcKoVo99c]. In conclusion Level III of the stochastic limit provides a clear illustration, in a multiplicity of fundamental physical models, of the basic philosophy of this theory namely: the physically interesting dilations of Markovian semigroups should be deduced from the basic Hamiltonian equations. The results emerged from the realization of this program show that the wealth and beauty of the structures hidden in the basic physical laws by far exceeds the fantasy displayed in the construction of phenomenological models. 6

Bibliography [AcLuVo00] Accardi L., Y.G. Lu, I. Volovich: Quantum Theory and its Stochastic Limit. to be published in the series Texts and monographs in Physics, Springer Verlag (2000); Japanese translation, Tokyo Springer (2000) [AcLuVo97c] Accardi L., Lu Y.G., I. Volovich The QED Hilbert module and Interacting Fock spaces. Publications of IIAS (Kyoto) (1997) [AcLuVo99] Accardi, L., Lu, Y.G., Volovich, I.V.: A white noise approach to classical and quantum stochastic calculus, Preprint of Centro Vito Volterra N.375, Rome, July 1999, World Scientific (2000) [Ac78] Accardi L.: On the quantum Feynmann-Kac formula. Rendiconti del seminario Matematico e Fisico, Milano 48 (1978) 135-180 [AcLu92] Accardi L., Lu Y.G.: The Wigner Semi circle Law in Quantum Electro Dynamics. Commun. Math. Phys., 180 (1996), 605 632. Volterra preprint N.126 (1992) [AcKo99a] L. Accardi, S.V. Kozyrev: The stochastic limit of quantum spin system. invited talk at the 3rd Tohwa International Meeting on Statistical Physics, Tohwa University, Fukuoka, Japan, November 8-11, 1999. to appear in the proceedings published by American Physical Society [AcKo99b] L. Accardi, S.V. Kozyrev: On the structure of Markov flows. to appear in: Chaos, Solitons and Fractals (2000) [AcKoVo98] L. Accardi, S.V. Kozyrev and I.V. Volovich: Dynamical origins of q-deformations in QED and the stochastic limit Journal of Physics A, Math. Gen. 32 (1999) 3485 3495 q-alg/9807137 [AcKoVo99c] L. Accardi, S.V. Kozyrev and I.V. Volovich: Non-Exponential Decay for Polaron Model Phys. Letters A 260 (1999) 31 38 [Ske96] Skeide M.: Hilbert modules in quantum electro dynamics and quantum probability. Volterra preprint N. 257 (1996). Comm. Math. Phys. (1998) 7

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Viacheslav P. Belavkin Quantum Stochastics as a Boundary Value Problem, and Classification of Quantum Noise. Abstract: Using a white (Poisson) analysis in Fock triples we formulate a class of boundary value problems for quantized fields which interact with a quantum system at the boundary. We prove that in the interaction representation the scattered fields plus the boundary satisfy a quantum stochastic equation for the Markovian unitary evolution in Fock space with respect to a quantum Poisson noise as an ultrarelativistic limit of the input fields. We give the complete classification of quantum noises as stochastic processes with independent increments indexed by arbitrary non-abelian Itô algebra, and prove that each such process can be decomposed into the orthogonal sum of quantum independent Brownian and Lévy motions. Every quantum stochastic unitary evolution driven by such noise corresponds to a unique self-adjoint boundary value problem for a free quantum field with the singular boundary interaction. Howard Carmichael Physical principles of quantum trajectories. Abstract: The earliest proposal of a stochastic evolution in quantum optics was that of Einstein, who put forward his so-called A and B theory to account for the approach to equilibrium of matter in interaction with black body radiation. Modern quantum trajectory methods are close relatives of the Einstein proposal. In this talk I trace the differences between quantum trajectories and the Einstein stochastic process, and the physical reasons that they must be introduced. The logic of a quantum trajectory as a conditioned evolution is set out and illustrated by a number of examples from quantum optics. The physical meaning of the term quantum jump is explored in relation to ongoing experiments in cavity QED. 22

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Inge S. Helland Experiments, symmetries and quantum mechanics. Looking at the development of quantitative methodology in this century, one of the most striking observation is that we throughout the entire period have had two different cultures, mathematical statistics and quantum theory, both working with prediction under uncertainty, but where there up to now has been practically no scientific contact between the two disciplines. One reason may be that the domains of application for the theories of the two disciplines are different, and there may possibly also be some real differences that can never be removed. Nevertheless, a systematic search for common ground may well be worth while, and may even lead to new insight on both sides. A very important issue will then be the language used in the various theories. We all know how efficient the use of probability theory and of the theory of decisions made from classes of probability measure is on the one side, and of the efficiency of concepts derived from functional analysis and operator theory on the other side. However, to be able to find a common ground, I think that we must try if possible to simplify concepts, even to such an extent that we in principle should be able to explain the conceptual foundation to lay persons. I feel that in the end, nobody could claim to have real understanding of a subject without having addressed such an aim. This does of course not mean that mathematics is unimportant. On the contrary, strong mathematics will always be very important in developing theories. But the basic foundation should be simple - at least ideally. Consider first the concept of experiment from mathematical statistics. The formal structure found in textbooks is (X, F, {P θ ; θ Θ}), where X is the space of possible outcomes of the experiments, F is a σ-algebra (Boolean algebra) of subsets of X, and we then have a class of probability measures on (X, F), indexed by θ. By use of some time and patience, I think that the basic idea here can be conveyed in relatively simple terms: The possible outcomes of an experiment may always be assumed to belong to some given space; the purpose of the experiment is to get some information on an unknown parameter which can be taken as a label of the probability measures that together constitute the model, and so on. On the other hand: The formal concept above obviously lacks many of the features that people link to the concept of experiment : First of all the preparations: choice of treatments, blocking, randomization and so on; this is rather trivial and well appreciated. But even as a way to describe the outcome of real experiments, the formal concept is defective: In this formal world nothing unexpected can happen, but in the real world this happens quite often. At best we can look upon the formal structure as some simplified frame that is useful when handling certain questions in statistical decision theory. Next consider the concept of state of a system. In quantum mechanics this is of course a ray in a Hilbert space or a density matrix. But for a layman this must seem like a very strange concept, and in this case it indeed seems very difficult to explain the concept in simple terms. 33

One possible approach might be to make a comparison to a completely different area where the concept of state is also used: Look at a medical patient. The state of this patient can then in principle be defined as the collection of all results of all tests/experiments that can be performed with him, some of which may be mutually incompatible. A complication is that such experiments may have random measurement errors, so in the state definition we should in some way talk about ideal experiments. From a statistical point of view, an alternative might perhaps be several large sets of independent, identically distributed experiments on the same patient, but this will represent a further complication in several ways, so we will avoid that. An alternative that is important in the quantum framework, would have been to include the measurement aparatus in the modelling framework - leading to socalled generalized measurements. This was pointed out to me afterwards by Richard Gill, but was not included in the talk. However, the issue seems to be possible to address by extending the models discussed below. The main purpose of this talk, then, is to discuss simply to which extent one can pass from the rather straightforward state concept of the medical patient to a similar state concept in quantum mechanics. More precisely: Let A be a set of potential experiments E a =(X a, F a, {Pθ a a ; θ a Θ a })fora A. Define a proposition as P =(a, E a ), where E a F a. (Perform an experiment, then observe an event in it.) Finally, we can always define φ such that each θ a = θ a (φ). The state should somehow be determined by φ in such a way that all the probability measures Pθ> a a can be found from the state. The basic question is to what an extent one can pass from this setting to the Hilbert space setting. We look at two approaches, one based upon quantum lattice theory and one based upon group representation theory. In the last case we will also have to assume in addition that there exists a group G on Φ = {φ}. In the quantum lattice approach (Helland, 1999) we first have to order partially the propositions that we have just defined. For propositions from the same experiment the ordering is obvious; in general we say that P 1 P 2 if P a 1 θ a1 (φ) P a 2 θ a2 (φ) for all φ. Wehave to identify events P 1 and P 2 if both P 1 P 2 and P 2 P 1. Under some additional assumptions this partially ordered set of propositions will form an orthomodular, orthocomplemented lattice. Specifically, these assumptions turn out to be essentially: 1) If we define the orthogonal complement of a proposition by (a, E) =(a, E c ), then we demand that pairwise orthogonality (P i Pj for all i and j) should imply orthogonality in the sense: k P a i θ i (φ) (E i) 1, φ. i=1 A similar condition is fundamental in the axiom set of Mackey (1963). 2) If the supremum of a proposition set {P i } (corresponding parameters θ i ) exists, then 34

it will be a proposition (with parameter θ) such that {f : f(φ) = f(θ(φ))} i {f : f(φ) = f(θ i (φ))}. The main point now is that there exist deep theorems (Beltrametti and Cassinelli, 1981, and references there) to the following effect: Under the conditions above and some additional technical assumptions (atomicity, covering property, separability) a Hilbert space model of the quantum theory type for the propositions can be constructed. In the discrete case these additional assumptions are automatically satisfied, so if we combine with Gleason s Theorem we get the conclusion: In the case where all experiments are discrete, and the assumptions above hold, there is a complex, separable Hilbert space H 0 such that (assuming that the dimension of H 0 is 3) each proposition P =(a, E) can be associated uniquely with a projection operator Π a,e in H 0 in the sense that P a θ a(φ) (E) = tr(ρπ a,e), where ρ = ρ(φ) is a density operator. Results of this kind definitively give some information about the interpretation of the quantum mechanical state concept and about the possiblity of finding a more classical interpretation of quantum mechanics. However this approach also has weaknesses: 1) The technical conditions needed above are disturbing. 2) The approach gives no explicit construction of the projection operators and of the density matrices. The alternative, symmetry based approach seems to be an improvement to the quantum logic approach with regard to both these aspects. It also seems to tie up with some recent development in statistical methodology, related to model reduction under symmetry. The details of the method are rather complex however, and work is still being done on improving some of these technical points. The main idea is very simple, however: Group representation theory gives for free a vector space and operators on this vector space related to any given group. This is well known and used as a tool in several quantum mechanical calculations. However, in our setting we aim at being more fundamental and possibly base the construction of state vectors on this vector space. An interesting point is the following: The connection from the original group to a matrix group in the representation is a homomorphism. Homomorphisms also appear in statistical estimation theory when parametric functions are to be estimated. Bohr & Ulfbeck (1995) have formulated a symmetry based quantum theory, to some extent on qualitative considerations. There are quantitative details to fill out, and, in the spirit of the present talk, the possibility of a translation from that theory to ordinary statistical theory should be investigated. Here are some details of our approach: 35

1. Consider a model for a closed physical system of the type formulated above, where φ is a hyperparameter, but where only certain parametric functions can be estimated. 2. A representation of the underlying group G on the Hilbert space L 2 (Φ,ν), with ν being Haar measure, is given by U(g)f(φ) =f(g 1 φ). 3. If ψ is an invariantly estimable function of φ (i.e., ψ(gφ) isalwaysafunctionofψ(φ)), then V = {f : f(φ) = f(ψ(φ)) is an invariant space of the operators {U(g)} above. This creates a 1-1 orderpreserving correspondance between the invariantly estimable functions and certain invariant subspaces. One can also construct a correspondence between parameter values and vectors in the space, given by f φ1 (φ) =f 0 (g 1 φ)whenφ 1 = gφ 0. Here, f 0 and g 0 are fixed. 4. Corresponding roughly to model reduction under scarce data, one can regard a fixed irreducible invariant space H as the state space. 5. Fix a Aand a function q. Using the Fourier transform of group representations one can show that there is a unique selfadjoint operator A q a on L 2 (Φ,ν) such that f φ 1 (A q a f φ 1 )=q(θ a (φ 1 )) for all φ 1. 6. Apart from technicalities, the only essential thing that seems to be missing from this scheme to create quantum formalism, is to postulate that the transition probabilities are symmetric: P (u v) =P (v u). We hope to complete the paper on this last approach before too long. We are also working on large scale statistical methods that appear to have at least some (admittedly, at present rather weak) relationship to this framework. The hope in the really long term, however, is still that there is a feasible road to the unity of science - a science where formal constructions are welcome as tools for doing calculations, but where the conceptual foundation somehow may be understood in simple terms. References. Beltrametti, E.G. & G. Cassinelli (1981) The Logic of Quantum Mechanics. Addison- Wesley, London Bohr. A. & O. Ulfbeck (1995) Primary manifestation of symmetry. Origin of quantal indeterminacy. Rev. Mod. Phys. 67, 1-35. Helland, I.S. (1999) Quantum mechanics from symmetry and statistical modelling. Int. J. Theor. Phys. 38, 1851-1881 Mackey, G.W. (1963) The Mathematical Foundations of Quantum Mechanics. Benjamin, New York. 36

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Peter Høyer The Subgroup Problem via State Distinguishability. Abstract: All known quantum algorithms which run super-polynomially faster than the most efficient probabilistic classical algorithm solve special cases of what is called the Abelian Subgroup Problem. This general formulation includes Shor s algorithms for factoring and finding discrete logarithms. The Abelian subgroup problem has a simple and beautiful quantum solution, and we give a very brief review of it. The heart of the idea behind the Abelian solution is Fourier analysis on Abelian groups and thus it is natural to consider how Fourier analysis on noncommutative groups might help in solving the general hidden subgroup problem. This is so far the most studied approach for possible generalizations to non-commutative groups. It is the difficulties of Fourier analysis on non-commutative groups that makes this a challenging approach. The hidden subgroup problem might be most easily understood when casted as a statedistinguishability problem: We are given a source that emits a fixed state out of a set of possible non-orthogonal mixed states. Our problem is to determine which one of those possible states the source emits. We can request several copies of the state and we are allowed to perform measurements on the joint system. For the subgroup problem, the set of possible states contains one state for each subgroup, and the state emitted by the source is the state corresponding to the hidden subgroup. Determining which state the source emits also determines the hidden subgroup. From the outcomes of our measurements, we can consider probabilistic analysis like Bayesian analysis (to conclude that, say, certain states are more likely than others), or we can limit ourselves to analytical tools as used in search theory (to rule out those states for which the outcomes would not have been possible). We discuss and exemplify all of the ideas above. No prior knowledge about the noncommutative problem will be assumed. 48

Uffe Haagerup Spectra of random matrices and random operators on Hilbert spaces. Abstract: The eigenvalue distribution of selfadjoint random matrices were first studied by E. Wigner in 1955, where he proved the so-called semicircle law for the eigenvalue distribution of large symmetric random matrices A with independent entries a(i, j) fori j. In 1967 a similar analysis were carried out by Marchenko and Pastur for the asymptotic eigenvalue distribution of Wishart matrices, i.e. symmetric random matrices of the form S = B T B,(whereB is an m n random matrix with independent entries b(i, j) inthe limit where m and n both goes to infinity while their ratio m/n stays almost constant). In both cases further developments have been made by mathematical physicists and by probabilists, particularly the asymptotic behaviour of the largest and smallest eigenvalue have been studied by Tracy and Widom et. al. in the Wigner case and by Geman, Silverstein, Bai, Yin et. al. in the Wishart case. In the talk, I will give a survey of these result, and present some recent results by Steen Thorbjørnsen and the speaker, where the asymptotic behaviour of the upper and lower bound of the spectrum of certain random operators on an infinite dimensional Hilbert space is studied. The motivation for the latter research does not come from physics or probability theory, but from problems in pure mathematics, more precisely from K-theory of C -algebras. Göran Lindblad Gaussian maps and processes in quantum systems. Abstract: Quasifree maps on the CCR algebra can be used to construct models of relaxation processes, information transfer, measurement processes, and many related types of irreversible quantum dynamics. This set of models are simple and convenient, allowing rather explicit calculations in many cases; they are particularly useful in quantum optics problems. In this talk I will give one or two examples. One if them involves a family of quasifree quantum cloning maps. They have a partial order which allow us to discuss the minimal noise added in such copying of quantum information. Multiple clones are easily handled. The formalism is very close to that for quantum amplifiers in quantum optics. 49

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56 Elena R. Loubenets Continuous in Time Nondemolition Observation of an Open System. 1. Introduction The problem of description of continuous in time measurement of an open system was considered in the well known papers of A.Barchielli, V.P.Belavkin, E.B. Davies, L.Diosi, A.S.Holevo, M.Ozawa, A.Peres [2-18, 22,23]. The special case of continuous in time indirect measurement of an open system called a nondemolition case was introduced by V.P.Belavkin in [9-15]. The well known quantum filtering equation [9-15, 5] describes the posterior quantum stochastic evolution of an open system subjected to continuous in time nondemolition observation in the special nondemolition measurement model of the so called quantum stochastic mechanics. In this talk we present the approach [1,20] to the description of continuous in time indirect nondemolition observation of an open system which is based on the methods of quantum theory and the Schro dinger equation but not on the methods of quantum stochastic calculus. 2. The description of continuous in time direct observation of a quantum system Continuous in time direct measurement of a quantum observable is possible if and only if in the Heisenberg picture this observable is a nondemolition one, that is if it satisfies the conditions Aˆ ˆ H ( t) = A + H ( t), (1) [ Aˆ ( ), ˆ H t AH ( t1)] = 0, for t, t 1. The process of continuous in time direct measurement of a nondemolition observable Aˆ H ( s) until the moment t is described by a projection-valued measure ˆ (0, t] P ( ) on the product space ( 0, t] Ω of all sets of possible outcomes of continuous observation of Aˆ H ( s) at all moments of time 0 < s t until t. For different moments of time the introduced measures commute and are compatible. 3. The description of continuous in time indirect measurement of an open system Let in the interaction picture induced by the free dynamics of reservoir the unitary time evolution of the extended system on the Hilbert space H S H R be described by the unitary operator U ˆ ( t, s). We consider the case when the indirect observation of an open system is performed by means of continuous in time direct observation of a nondemolition observable Qˆ H ( t) of the reservoir modelling the measuring device (2) ˆ ˆ + Q ( ) (, 0)( ˆ ˆ( )) ˆ H t = U t I Q t U ( t,0). The POV measure of the indirect measurement is given by (0, ] (0, ] (0, ] (0, ] (0, ] (0, ] (3) ˆ t t ( ), ˆ t t t t M E =< ϕ P ( E ) ϕ >, E Ω R Qˆ H ( 0, t] on the product space Ω of all sets of possible outcomes of continuous in time indirect observation until the moment t. The description of the process of continuous in time indirect measurement of an open system is similar to the description of a classic stochastic process with a scalar probability measure R H R Aˆ H

57 (4) µ (0, t] ( E (0, t] ) ˆ (0, t] (0, t] = ψ 0, M ( E ) ψ 0 ( 0, t] on Ω induced by initial states ψ 0 and ϕ R of an open system and a reservoir, respectively. The consideration of the case when an open system and a reservoir are described initially by density operators is obvious. The family of POV measures (3) ( the family of scalar probability measures (4) as well) is compatible. 3. Nondemolition measurement Consider a special case of continuous in time indirect measurement called a nondemolition case. The main principles of nondemolition observation were introduced by V.P.Belavkin in [9-15] and imply that in the Heisenberg picture the continuously observed nondemolition observable Qˆ H ( t) commutes with all observables of an open system under s t. (5) [ Zˆ ( t), Qˆ ( s)] = 0 H H, In this representation we identify a nondemolition measurement as one defined by the conditions Qˆ( t) = Qˆ + ( t), [ Qˆ( t), Qˆ ( t1)] = 0, for t, t1. (6) [ Uˆ ( t, s),( Iˆ Qˆ( r))] = 0, r < s t, which are sufficient for a continuous in time indirect measurement to be a nondemolition one according to V.P.Belavkin's definition. The families of nondemolition observables { Qˆ( s),0 < s t} and { Qˆ H ( s), 0 < s t} are usually called [9-15] as input and output operatorvalued processes, respectively. In the case of continuous in time nondemolition measurement we derive: ˆ (0, t] (0, t] ˆ + ( ) (,0)( ˆ ˆ (0, t] (0, t ] P ( )) ˆ ˆ E = U t1 I Pˆ E U ( t1,0), Q (7) H Q (0, t] (0, t] for E Ω, t1 t, where ˆ (0, t] P Q ( ) is a projection -valued measure describing the process of continuous in time direct measurement of the input nondemolition observable Q ˆ( t ) until the moment t. Consequently, the POV measure ˆ (0, t] M ( ) given by (3) in the nondemolition case has the form: ˆ (0, t] (0, t ] ˆ + ( ), (,0)( ˆ ˆ (0, t] (0, t ] M E = ϕ ( )) ˆ R U t I Pˆ E U ( t,0) ϕ R, Q (8) H R (0, t] (0, t ] E Ω. For simplicity, suppose the input observable Q ˆ( t ) is continuous with respect to t in the strong operator topology and for any moment t the family of commuting operators { Qˆ( s)} s (0, t ] on H R is full. Denote by Q( s) R a possible outcome of a direct measurement of ˆ t Q H ( s) at the moment s. A time-ordered sequence Q = { Q( s) } s (0, t ] of possible outcomes of continuous measurement of ˆ ( s) at all moments of time until t is an element of Q H ( 0, t] Ω. Due to the properties of a projection-valued measure corresponding to a self-adjoint operator, a space Ω (s) of possible outcomes of measurement at the moment s can be ( 0, t] identified with R. Consequently, the product space Ω can be identified with a space of H S, E (0, t] Ω (0, t]

trajectories. Due to our assumption of continuity of the operator Q ˆ( t ), ( 0, t ] Ω is a space of trajectories "continuous" with respect to t in the sense of the scalar probability measure induced by the input process (0, t] (0, t] (9) ( ), ˆ (0, t ] (0, t] ν ˆ E =< ϕ R Pˆ ( E ) ϕ R >. Q At any moment of t the following resolution of identity on the Hilbert space H R of a reservoir is valid (10) ˆ ˆ (0, t ] t I = P Q ( dq ). t (0, t ] Q Ω Q ˆ t Introduce a linear operator Vˆ[ Q, t; ϕ R ] on the Hilbert space H S of an open system by the following relation ˆ t ˆ (0, t] t ˆ ˆ (0, t] t ( V [ Q, t; ϕ ] ( ))( ) ( ( )) ˆ R Pˆ dq ψ ϕ R = I Pˆ dq U ( t,0)( ψ ϕ R ), Q Q (11) for ψ H S, understood in the infinitesimal sense. The POV measure, defined by (8), is (12) ˆ (0, t] (0, t ] ˆ + t t (0, t] t (0, t] (0, t M ( E ) = V [ Q, t; ϕ ] Vˆ[ Q, t; ϕ ] ν ( dq ), for E Ω ]. t ((0, t ] Q E R (0, t] The scalar probability measure µ ( ) of the output process is given through a scalar probability measure of the input process by (0, t] (0, t] (13) ˆ + t t (0, t] t µ ( E ) = ψ 0, V [ Q, t; ϕ R ] Vˆ[ Q, t; ϕ R ] ψ 0 ν ˆ ( dq ). Q t (0, t ] Q E From the definition (11) the following representations follow (14a) t t t Uˆ t ψ ϕ = V Q t ϕ Pˆ (0, ] (,0)( ) ( ˆ[, ; ] ( dq )) ( ψ ϕ ), (14b) R t (0, t ] Q Ω R < ˆ t (0, t] t ϕ, U ( t,0) ϕ > = Vˆ[ Q, t; ϕ ] ν ( dq ). R R H R t (0, t ] Q Ω Qˆ R Qˆ R Qˆ R H S for ψ H t The operator Vˆ[ Q, t; ϕ R ] describes the irreversible in time stochastic evolution of an open system under the condition that the output process Qˆ H ( s) was continuously observed until t the moment t and found to have the trajectory Q. The state t t χ[ Q, t; ϕ R ] = Vˆ[ Q, t; ϕ R ] ψ 0, (15) for ψ 0 H S can be interpreted as the posterior state (unnormalized) of an open system subjected to continuous in time nondemolition observation In the general case of nondemolition measurement, considering the Cauchy problem for the operator U ˆ ( t,0), we derive the following integral equation for the posterior state: t t i ˆ τ χ[ Q, t; ϕ R ] = ψ 0 + H S χ[ Q, τ ; ϕ R ] dτ + 0 (16) t i ˆ τ τ τ (0, τ ] τ + W[ Q, Q1, τ; ϕ R ] χ[ Q1, τ; ϕ R ] ν ˆ ( dq1 ) dτ. Q 0 τ (0, t ] Q 1 Ω where S 58

59 (17) ( Wˆ [ Q t, Q t 1, t; ϕ R ] ν (0, t] Qˆ ( dq t 1 )) Pˆ (0, t] Qˆ ( dq t )( ψ ϕ = ( Iˆ Pˆ (0, t] Qˆ R ) = ( dq t )) Hˆ int ( t)( Iˆ Pˆ (0, t] Qˆ ( dq t 1 ))( ψ ϕ for ψ H S, and Hˆ, ˆ S H int ( t) are the Hamiltonians of an open system and of the interaction (in the interaction picture), respectively. In the general case of nondemolition measurement (18) ˆ t t ˆ t t (0, t) (0, t W[ Q, Q, ; ] [,, ; ] ( ) 1 t ϕ R = W1 Q Q1 t ϕ R δν Q1 Q ), so that the third term in (16) can be essentially simplified. The notion of the posterior state of an open system under continuous in time observation was first introduced by V.P.Belavkin in [9-15 ] in the special nondemolition measurement model of quantum stochastic calculus. The notion similar to the notion of the quantum stochastic t evolution operator Vˆ[ Q, t; ϕ R ] was introduced in [9-15] through the concept of the generating map of an instrument [2] and defined in the integral form, so that the derivation of the posterior state equation based on the use of quantum stochastic calculus is rather complicated. The equation for the posterior state derived in [9-15] is valid only for the case of the nondemolition measurement model of quantum stochastic calculus. Our approach to the description of continuous in time indirect measurement, our definition t (11) of the quantum stochastic evolution operator Vˆ[ Q, t, ϕ R ] allow us to derive the integral equation (16) describing the posterior dynamics of an open system under continuous in time observation in the general nondemolition case. 5. The special nondemolition measurement model of quantum stochastic calculus, the extended variant In the case of the special nondemolition measurement model of quantum stochastic calculus the unitary time evolution Uˆ ( t,0) Uˆ ( t) of the extended system in the interaction picture is described by the quantum stochastic differential equation (19) ˆ ˆ ˆ+ ˆ ˆ + du ( t) = ( K Iˆ) dt ( L Rˆ) daˆ( t) + L da ( t) + ( Rˆ Iˆ) dλˆ ( t) ) Uˆ ( t), introduced by Hudson R.L. and Parthasarathy K.R. [19]. In (19) Lˆ R Kˆ Kˆ Lˆ + ; ˆ; + + = Lˆ are operators on the Hilbert space H S of an open system, Rˆ is unitary. The exact relation between the operator Kˆ in (19) and the Hamiltonian open system in the case when Rˆ Iˆ is given in [20]: i (20) Kˆ H Lˆ + R Rˆ 1 = + ˆ( + Iˆ) Lˆ. S Ĥ S of an The reservoir, modelling the measuring device, is described as a Bose field and is represented by a symmetric Fock space over a single particle Hilbert space Z which in applications in quantum stochastic mechanics is taken to be L 2 ( R). The annigilation, creation and gauge (or number) operators ˆ ˆ + A( t), A ( t), ˆ ˆ + Λ ( t) = Λ ( t) describing the free dynamics of the Bose field form the annigilation, creation and gauge operator-valued processes, respectively. Denote by e( f ), f L 2 ( R) -- a normalized coherent vector for the Bose field: t ~ (21) Aˆ( t) e( f ) = ( f ( τ ) dτ ) e( f ), 0 ~ where f ( t ) is a Fourier transform of a function f L 2 ( R ).Considering the Hamiltonian [20] of the Schr o dinger equation corresponding (when Rˆ I ˆ ) to the quantum stochastic R ),

60 differential equation of Hudson- Parthasarathy, we derive the following equation for the posterior state of an open system in the case of diffusion observation and the initial state of the reservoir ϕ = e( f R ) : (22a) t χ[ Q, t; e( f )] = ψ 0 t 0 i ( Hˆ where we introduce (22b) ˆ 1 ˆ2 ˆ+ G( t) = L + L Rˆ( Rˆ + Iˆ) 2 + 1 S t 0 Lˆ + Gˆ ( τ )) χ[ Q τ, τ; e( f )] dτ + ~ q( τ ){ Lˆ + f ( τ )( Rˆ Iˆ)} χ[ Q + 1 2 ~ f ( t)( Rˆ Iˆ) Lˆ + + 1 2 τ, τ; e( f )] dτ, ~ ~ ( ) ˆ( ˆ ˆ) ( ) ˆ+ f t L R + I + f t L Rˆ + 1 2 ψ ~ 2 f ( t)( Rˆ 2 0 H Iˆ), dq( t) and q( t) = - the generalized derivative of the function Q (t). In the considered case of dt diffusion observation, q (t) is a piecewise continuous function. Using the methods of the classic stochastic calculus, we can rewrite the integral equation (22) in the stochastic form: t χ[ Q, t; e( f )] = ψ (23) t 0 0 i { Hˆ S ˆ+ + L Rˆ( Rˆ + Iˆ) + t 0 1 ˆ ~ ˆ+ ˆ ~ 2 ˆ) ( )( ˆ τ L + f ( τ )( L R + L + f τ R Iˆ)} χ[ Q, τ; e( f )] dτ + ~ { Lˆ + f ( τ )( Rˆ Iˆ)} χ[ Q τ, τ; e( f )] dq( τ ), ψ 0 H The stochastic integral in (23) is understood in Ito's sense, dq (t) is a stochastic differential of the classic diffusion process corresponding to the continuous in time direct observation, when ϕ = e( f R ), of the input nondemolition observable Q ˆ( t) = Aˆ + ( t) + A ˆ( t). The scalar probability measure of the input diffusion process is defined by (9). We would like to mention that the stochastic integral equation (23) can be also derived by substituting U ˆ ( t,0), defined by (19), directly into the definition (11). The stochastic integral equation (23) for the posterior state of an open system under continuous in time observation of diffusion type coincides with the quantum filtering equation for the case of diffusion observation [15] only if the initial state of a reservoir is vacuum and Rˆ = Iˆ. 3. Concluding remarks We present an approach to the description of continuous in time indirect measurement which allows us: - to derive the new results in the general case of nondemolition measurement; - to derive the new integral equation for the posterior state of an open system in the special nondemolition measurement model which is the extended variant (including the gaude term) usually considered in quantum stochastic mechanics. S S,.

61 REFERENCES [1] A berg, C., Loubenets, E.R. (1994). Quantum stochastic calculus and nondemolition measurements. Notes on a recent approach to quantum measurements. Report No 94-28, 48pp., Institute of Theoretical Physics, G o teborg, Sweden. [2] Barchielli, A., Lanz, L., Prosperi, G.M. (1983). Statistic of continuous trajectories in quantum mechanics: operation valued stochastic processes. Found.Phys, 13, 79-812. [3] Barchielli, A. (1985).Quantum stochastic calculus, operation-valued stochastic processes and continual measurements in quantum mechanics. J.Math.Phys. 26, 2222-2230. [4] Barchielli, A. (1986). Measurement theory and stochastic differential equations in quantum mechanics. Phys. Rev. A, 34, 1642-1648. [5] Barchielli, A.. Belavkin, V.P. (1991).Measurements continuous in time and a posteriori states in quantum mechanics.j.phys.a; Math.Gen. 24,1495-1514. [6] Barchielli, A. (1993).On the quantum theory of measurements continuous in time. Reports on Math.Phys. 33, 1/2, 21-34. [7] Barchielli, A., Holevo, A.S. (1995). Constructing quantum measurement processes via classical stochastic calculus. Stochastic Processes Appl. 58, 293-317 [8] Barchielli, A. (1998). On stochastic differential equations and semigroups of probability operators in quantum probability. Stoch. Proc. and their Applications 73, 69-86. [9] Belavkin, V.P. (1988). Nondemolition measurements, nonlinear filtering and dynamic programming of quantum stochastic processes. In: Blaquiere, A. (Ed.), Lecture Notes in Control and Information Sciences 121, 245-265, Springer, Berlin. [10] Belavkin, V.P. (1989). Stochastic calculus of input-output processes and nondemolition filtering. VINITI 36, 29-67, Moscow. [11] Belavkin, V.P. (1989). A continuous counting observation and posterior quantum dynamics. J. Phys.A 22, L1109-L1114. [12] Belavkin, V.P. (1989). A new wave equation for a continuous nondemolition measurement. Phys. Lett. A 140, 355-358. [13] Belavkin, V.P. (1990). Stochastic equations of quantum filtering. Proceedings of the Fifth Int.Conference "Probab.Theory and Math.Statistic", Vilnus. [14] Belavkin, V.P.(1992). Quantum continual measurements and a posteriori collapse on CCR. Commun. Math.Phys.146, 611-635. [15] Belavkin, V.P. and Staszewski, P. (1992). Nondemolition observation of a free quantum particle Phys. Rev. A, 45, 1347-1356. [16] Davies, E. B. (1976). Quantum Theory of Open Systems. Academic Press, London. [17] Diosi, L.(1988). Continuous quantum measurement and Ito formalism. Phys.Lett.A 129, 419-423. [18] Holevo, A.S. (1987), Quantum estimation. Advances in Statistical Signal Processing 1, 157-202, JAI Press Inc. [19] Hudson, R.L., Parthasarathy, K.R. (1984). Quantum Ito's formula and stochastic evolutions. Comm. in Math. Phys., 93, 301-323. [20] Loubenets, E.R. (1999).Quantum stochastic evolutions and the Schro dinger equation. Russian Journal of Mathematical Physics. (To appear). [21] Loubenets, E.R. (1999). Quantum stochastic evolution of an open system under continuous in time nondemolition measurement. Research Report No. 45, Centre for Mathematical Physics and Stochastics, University of Aarhus, Denmark. [22] Ozawa, M.(1884). Quantum measuring processes of continuous observables. J.Math.Phys.25, 79-87. [23] Peres, A. (1993). Quantum Theory: Concepts and Methods. Kluwer, Dordrecht.

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Günther Mahler Statistical measures for the characterization of quantum trajectories. Abstract: It has been a widespread believe that the notion of a trajectory has to be abandoned in quantum mechanics. This certainly holds for the classical phase-space; nevertheless, the unitary motion of the state vector in Hilbert space may well be interpreted as generating an abstract trajectory. For open quantum systems, though, such a concept remained obscure until the advent of stochastic unravelling techniques of the underlying master-equation. While there is substantial progress in the description of non-classical states, the notion of non-classical trajectories has received much less attention. For stationary models we consider the statistics of 4 different features: (i) coherence measures (distributions), (ii) uncertainty of relevant observables (variances), (iii) information aspects (entropies), and (iv) jump distances (pertinent distributions). Typical results are illustrated by means of numerical simulations. Serge Massar How much classical communication is required to simulate quantum communication and quantum entanglement?. Abstract: I will analyze how much classical communication is required to simulate quantum communication and quantum entanglement. I first show that it is impossible to classically simulate quantum communication and entanglement by exchanging only a finite number of classical bits. However if the two parties share an infinite number of random bits (ie. an infinite number of local hidden variables) then simulation of quantum communication and entanglement is possible with only a finite number of bits of communication, as shown recently by Brassard, Cleve and Tapp. Furthermore even if the two parties do not share an infinite number of random bits, simulation of quantum communication and entanglement is possible with an amount of communication which is finite in the mean. 72

Ian Percival Generalized Bell inequalities. 1 Introduction The original Bell inequality is one example of a wide class of inequalities that relate the conditional probabilities of classical events in spacetime. The inequalities can be expressed in a beautiful geometrical form, which is a many-dimensional generalization of a threedimensional problem in diamond-cutting. A prominent diamond-cutter from Amsterdam has a rich and eccentric customer, who has given precise specifications for a diamond, in terms of the coordinates of each vertex. But the cutting of diamonds is not defined in terms of the vertexes, it is defined by the planes of the cutter, which follow the plane surfaces or facets of the diamond. So the diamondcutter needs the equations of the facets, not the coordinates of the vertexes. The problem of finding all the generalized Bell inequalities is like the problem of the diamond-cutter, extended to arbitrarily high dimensions. The shape of a diamond of arbitrarily high dimension is called a polytope, and the space occupied by the diamond is the convex hull of its vertexes. General Bell inequalities are of interest to physicists and philosophers of science, but their presentation is particularly simple in the language of the engineer s input-output systems, particularly stochastic systems, which require the use of probability theory. A summary of the theory of transfer functions for stochastic systems in spacetime is given in Percival (1999). It expands on sections 2-4 of this abstract, with illustrative figures. 2 Cause and effect Traditionally causes come before effects, but according to special relativity, things are not that simple, and the Bell inequalities show that for interaction between classical and quantum systems, the relation between cause and effect in spacetime is even more subtle. 3 Deterministic systems Our systems are like the engineer s systems, with discrete classical inputs i and outputs j, but unlike the engineer s systems, which occupy a given region of space, ours occupy a region of spacetime. The input and output ports occupy restricted regions of spacetime, which for our purposes may be considered as points. The relations between an input and an output depends on whether they are separated by a spacelike interval or a timelike interval, and for the latter on the sign of the time. For classical deterministic and stochastic systems the input can only affect the output if the output port is in the forward light cone of the input port. The inputs and outputs may be connected by classical systems, as for the gates of classical computers, or by quantum systems, as for experiments to test Bell s inequality. For such experiments the causal relations between input and output in spacetime are not so simple. 73

Suppose that for a system with one input and one output port the number of possible inputs is N(i) and the number of possible outputs is N(j). Then there are N(i) N(j) possible transitions between the two. A particular deterministic system is defined by all the transitions (i j), one from each input. This is the transition picture. These transitions can be considered together as a functional relation between the input and the output, given by a transfer function F,where j = F (i). (1) This is the transfer picture. The number of possible transfer functions is N(F )=N(j) N(i). (2) There is a trivial relation between the transition and transfer pictures for deterministic systems, but not for stochastic systems. Now suppose we have two subsystems A and B of a system A+B that do not interact with one another. The constraint of no interactions can be expressed in terms of transfer functions. We denote the inputs, outputs and transfer functions for the subsystems by suffixes A and B. For deterministic systems, this independence can be expressed in terms of transfer functions. The subsystems A and B have separate transfer functions F A,F B such that j A = F A (i A ), j B = F B (i B ). (3) The transfer function for the whole system is F,where (j A,j B )=F (i A,i B ) where F =(F A,F B ) (4) This is the independence constraint on the transfer function F for the whole system. The number of transfer functions for these two independent systems is much less than the number of transfer functions for an arbitrary system with two inputs and two outputs of thesametype. 4 Stochastic systems and Bell experiments In the transition picture the relation between the input and the output is uniquely defined in terms of the transition or conditional probabilities P (j i) =P (i j) (0 P (i j) 1, P (i j) =1). (5) In the transfer picture the behaviour of the stochastic system is defined in terms of the probabilities P (F ) that the stochastic system behaves like a deterministic system with transfer function F. The relation between the two pictures is given by j P (i j) = F P (F )δ(j, F (i)). (6) In a standard Bell experiment, two photons with total spin zero are produced by downconversion, and travel in opposite directions to distant locations labelled A and B, where 74

their polarizations are measured at A and B by using a polarizing beam splitter and detectors. The angle θ A of the beam-splitter determines the direction in which the polarization is measured, with two possible output values j A. The same applies at B with an angle θ B and two possible values for the output j B. For different experiments there are a different possible values of the angles, which are labelled by the input values i A,i B.There are a finite number of possible values for these integer variables, typically two or three, depending on the type of experiment. The settings of the angles and the recordings of the polarizations are the classical events. The whole experiment may be considered as a single system, with input i =(i A,i B )and output j =(j A,j B ), or as a pair of subsystems A and B. To test locality in this experiment, it is essential that both the input and output ports (events) at A should be separated by spatial spacetime intervals from both the input and output events at B. In that case, according to special relativity, no signal can be sent from an input or output of A to an output of B, nor from an input or output of B to an output of A. For stochastic systems this can be expressed in terms of probabilities of transfer functions: P (F ) = 0 unless F =(F A,F B ). (7) This does not mean that the outputs for systems A and B are uncorrelated. The correlation can come about because P (F A,F B ) P (F A )P (F B ), (8) which is not excluded, because the transfer functions can be correlated through interactions in their common past. Now look at equation (6). For a given set of transition probabilities, and no constraint on the transfer function probabilities, it is always possible to find P (F ) to satisfy this equation. Usually there are many solutions. But as we remove transfer functions one by one, setting the corresponding P (F ) to zero, (6) becomes more and more difficult to satisfy, until a point is reached for which there are P (i j) that satisfy the conditions of (5), but cannot be expressed in the form (6). For the constraint (7), these are the transition probabilities that violate the Bell inequalities. A more detailed account of transfer functions for deterministic and stochastic systems, with a derivation of the simplest Bell inequality in terms of transition probabilities can be found on the lanl server: quant-ph/9906005. General Bell inequalities Now we are ready to generalize. Suppose there are N(s) subsystems, with all the inputs and outputs of each subsystem spatially separated from all the inputs and outputs of all the other subsystems. Suppose each subsystem has N (i) inputs and N (j) outputs, so that the whole system has N(i) =N(s)N (i) and N(j) =N(s)N (j) (9) inputs and outputs. To analyse these systems, we work in the space of the transition probabilities P (i j) of 75

the whole system. There are N(i)N(j) such transition probabilities, of which N(i)(N(j) 1) are independent, due to the summation over the output probabilities for a given input adding to unity. A deterministic system is a special example of stochastic system in which all the transition probabilities are either zero or one. The values are determined by the transfer function F. Now look again at equation (6). The delta function can be considered as the coordinates of the transfer function F in the space of transition probabilities. The whole equation expresses the coordinates of the transition probabilities as a weighted mean over the coordinates of the transfer functions. Clearly if there are constraints on the transfer functions then the weights for the forbidden transfer functions are zero and for the allowed transfer functions can be nonzero. The geometric interpretation of this weighted mean is that a point which represents an allowed set of conditional probabilities must lie in the convex hull of the points which represent the allowed transfer functions. This convex hull is the allowed polytope, a many-dimensional generalization of the three-dimensional diamond of the first section. The allowed sets of conditional probabilities lie inside this polytope. Each facet of the polytope defines an independent linear inequality that has to be satisfied by the point representing the conditional probabilities. These are the generalized Bell inequalities. By special relativity, if all the subsystems are linked classically, then subsystems must be independent in the sense of having separate transfer functions, and the transition probabilities must satisfy these generalized Bell inequalities. The remarkable thing about systems which are linked quantum-mechanically is that the transition probabilities need not satisfy the generalized Bell inequalities. Applications Generalized Bell inequalities can be applied to experiments with quantum systems with higher spin, as has been done by Garg and Mermin. They can also be applied directly to the raw data of imperfect experiments, which is important, because all real experiments are imperfect. For example in the case of the original Bell experiment for which N(s) =2, a perfect experiment would have N (j) = 2 corresponding to the two orthogonal states of polarization. But detectors are far from perfect, and often fail to count the photons. In addition, the photons may be absorbed before they reach the detectors. So there are three possible outputs for each subsystem, corresponding to the two polarizations of one photon, and zero photons. So instead of N(j) = 4 for the perfect experiment, we have N(j) = 9, and generalized Bell inequalities are required. The generalized inequalities are also needed for planning future experiments of a more general kind. There is an extensive literature on the subject of general Bell inequalities, of which the most accessible for physicists is the recent paper of Peres. Those more familiar with stochastic terminology and methods may prefer the book of Deza and Laurent. 76

References for physicists J.S.Bell 1987 Speakable and unspeakable in quantum mechanics. Cambridge University Press. A.Peres 1995 Quantum theory: concepts and methods. Kluwer, Dordrecht. E.P.Wigner 1970 On hidden variables and quantum mechanical probabilities. Am. J. Phys. 38 1005-1009. J.F.Clauser and A. Shimony 1978 Bell s theorem: experimental tests and implications. Rep. Prog. Phys. 41 1881-1192. A.Garg and N.D.Mermin 1984 Farkas lemma and the nature of reality. Found. Phys. 14 1-39. A.Garg and N.D.Mermin 1987 Detector inefficiencies in the Einstein-Podolsky-Rosen experiment. Phys. Rev. D 35 3831-3835. A.Peres 1999 All the Bell inequalities. Found. Phys. 29 589-614. I.C.Percival 1999 Quantum measurement breaks Lorentz symmetry. LANL quant-ph/9906005 References for others I.Pitowsky 1989 Quantum probability - quantum logic. Springer Lect. Notes on Physics 321, Springer, Berlin M.M.Deza and M.Laurent 1997 Geometry of cuts and metrics. Springer, Berlin. 77

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Relative entropy in quantum information theory Dénes Petz Mathematics Institute Technical University of Budapest H-1521 Budapest XI. Sztoczek u. 2, Hungary 1 Relative entropy and its properties The relative entropy is an information measure representing the uncertainty of a state with respect to another state. In the quantum (or noncommutative) case, states correspond to positive normalized functionals on an operator algebra and they are often given by density operators. When densities are available, the entropy of ω with respect to ϕ is defined by S(ω, ϕ) ={ Tr Dω (log D ω log D ϕ ) if supp D ϕ supp D ω + otherwise. This formula was given by Umegaki as a generalization of the Kullback-Leibler information for discrimination [18]. The relative entropy may be defined for linear functionals of an arbitrary C*-algebra. The general definition due to Araki goes through von Neumann algebras and is based on the concept of relative modular operator, see [1]. An equivalent definition works for nuclear algebras: S(ω, ϕ) =sup{s(ω α, ϕ α) :α}, where sup is taken over all unital Schwarz maps α : A 0 Asuch that A 0 is of finite dimension. (For the states ω α and ϕ α densities are available and the previous definition applies.) Theorem 1.1 The relative entropy of positive functionals of a C*-algebra posesses the following properties. (1a) (ω, ϕ) S(ω, ϕ) is convex and lower semicontinuous. (1b) ϕ ω 2 2S(ω, ϕ) if ϕ(i) =ω(i) =1. (1c) S(ω, ϕ 1 ) S(ω, ϕ 2 ) if ϕ 1 ϕ 2. (1d) The relation S(ω α, ϕ α) S(ω, ϕ) holds for a unital Schwarz map α : A 0 A. (1e) S(ω, ϕ)+ n i=1 S(ω i,ω)= n i=1 S(ω i,ϕ) for ω = i ω i. Partially supported by Hungarian National Science Foundation Grant OTKA F023447 84

These properties have been obtained in different papers at different levels of generality. (1a) is related to [1] and [10], (1b) is essentially from [6], (1d) is a result from [17] and (1e) is from [2] and [4]. Now we list the properties of the relative entropy functional which were used in an axiomatic characterization in [14]: (2a) Conditional expectation property: Assume that A is a subalgebra of B and there exists a projection of norm one E of B onto A such that ϕ E = ϕ. Then for every state ω of B S(ω, ϕ) =S(ω A,ϕ A)+S(ω, ω E) holds. (2b) Monotonicity property: For every unital Schwarz mapping α we have S(ω, ϕ) S(ω α, ϕ α). (2c) Direct sum property: Assume that B = B 1 B 2. Let ϕ 12 (a b) =λϕ 1 (a) +(1 λ)ϕ 2 (b) andω 12 (a b) = λω 1 (a)+(1 λ)ω 2 (b) for every a B 1,b B 2 and some 0 <λ<1. Then S(ω 12,ϕ 12 )= λs(ω 1,ϕ 1 )+(1 λ)s(ω 2,ϕ 2 ). (2d) Lower semicontinuity property: The function (ω, ϕ) S(ω, ϕ) is lower semicontinuous on the state space of a C*- algebra. Theorem 1.2 If an extended positive valued functional R(ω, ϕ) is defined for states of nuclear C*-algebras such that it has properties (2a) (2d), then there exists a constant c R {+ } such that R(ω, ϕ) =cs(ω, ϕ). The von Neumann entropy of a state of a finite quantum system is the relative entropy with respect to the tracial state, at least up to a sign and an additive constant: S(ω) = Tr (D ω log D ω ) Hence several properties of the von Neumann entropy follow from those of the relative entropy. 2 Model for quantum communication Let α : A Bbe positive unitial mapping and ϕ be a state of B. Such α may be interpreted as a channel in the sense that a state ϕ of B is transformed into another state ϕ α of A. ϕ is the initial or input state and ϕ α is the corresponding output state. It is always convenient to assume that α is not only a positive mapping in the usual sense but completely positive. To show a concrete example, consider the Stokes parametrization of 2 2 density matrices. D x = 1 2 (I + x 1σ 1 + x 2 σ 2 + x 3 σ 3 ), where σ 1,σ 2,σ 3 are the Pauli matrices and (x 1,x 2,x 3 ) R 3 with x 2 1 +x2 2 +x2 3 1. For a positive semi-definite 3 3 matrix A the application Γ : D x D Ax gives a channeling transformation when A 1. This channel was introduced in [5] under the name of symmetric binary quantum channel. (In order to have the dual of a completely positive mapping we need to impose some conditions on the eigenvalues of A.) 85

The quantum mutual entropy is defined after [11] as I(ϕ; α) =sup{ j λ js(ϕ j α, ϕ α) : j λ jϕ j = ϕ}, (2.6) where the least upper bound is over all (orthogonal) extremal decompositions. The purely quantum capacity C(α) ofα is the least upper bound of the mutual information quantities I(ϕ; α), where ϕ is varying over a set of states, possibly over all states. In order to estimate the quantum mutual information, we introduce the concept of divergence center. Let{ω i : i I} be a family of states and R>0. We say that the state ω is a divergence center for {ω i : i I} with radius R if S(ω i,ω) R for every i I. In our discussion about the geometry of relative entropy (or divergence as it is called in information theory) the ideas of [3] can be recognized very well. Let {ω i : i I} be a family of state. We say that the state ω is an exact divergence center with radius R if R =inf ϕ sup {S(ω i,ϕ)} i and ω is a minimizer for the right hand side. (When R is finite, then there exists a minimizer, because ϕ sup{s(ω i,ϕ) : i I} is lower semicontinuous with compact level sets, cf. Proposition 5.27 in [12].) The following result is from [13]. Theorem 2.1 Let Λ be a channeling transformation sending density matrices on the Hilbert space H to those of the other Hilbert space K. Assume that K is finite dimensional. Then the capacity of Λ is the divergence radius of the range of Λ. We want to compute the capacity of the above Γ. Since a unitary conjugation does not change capacity obviously, we may assume that A is diagonal with eigenvalues 1 λ 1 λ 2 λ 3 0. The range of Γ is visualized as an ellipsoid with (Euclidean) diameter 2λ 1.Itisnot difficult to see that the tracial state τ is the exact divergence center of the segment connected the states (I ± λ 1 σ 1 )/2 and hence τ must be the divergence center of the whole range. The divergence radius is ( ( ) 1 1 0 S + λ ( ) ) 1 0,τ 2 0 0 2 0 1 ( ( )) 1 1+λ 0 =log2 S =log2 η((1 + λ)/2) η((1 λ)/2). 2 0 1 λ This gives the capacity according to the previous theorem. 3 An infinite system setting of the source Let X n denote the set of all messages of length n. If x n X n is a message then a quantum state ϕ(x n )ofthen-fold quantum system is corresponded with it. The Hilbert space of the n-fold system is the n-fold tensor product H n and ϕ(x n ) has a statistical operator D(x n ). If messages of length n are to be transmitted then our quantum source should be put in the mixed state ϕ n = x n p(x n )ϕ(x n ) with statistical operator D n = x n p(x n )D(x n ), where 86

p(x n ) is the probability of the message x n. Since we want to let n, it is reasonable to view all the n-fold systems as subsystem of an infinite one. In this way we can conveniently use a formalism standard in statistical physics. Let an infinitely extended system be considered over the lattice Z of integers. The observables confined to a lattice site k Z form the selfadjoint part of a finite dimensional matrix algebra A k, that is the set of all operators acting on the finite dimensional space H. It is assumed that the local observables in any finite subset Λ Z are those of the finite quantum system A Λ = n i=1 A k. k Λ The quasilocal algebra A is the norm completion of the normed algebra A = Λ A Λ,the union of all local algebras A Λ associated with finite intervals Λ Z ν. A state ϕ of the infinite system is a positive normalized functional A C. It does not make sense to associate a statistical operator to a state of the infinite system in general. However, ϕ restricted to a finite dimensional local algebra A Λ admits a density matrix D Λ. We regard the algebra A [1,N] as the set of all operators acting on the N-fold tensor product space H N. Moreover, we assume that the density D N from the first part of this section is identical with D [1,N]. Under this assumptions we call the state ϕ the state of the (infinite) channel. Roughly speaking, all the states used in the transmission of messages of length n are marginals of this ϕ. Coding, transmission and decoding could be well formulated using the states ϕ N ϕ [1,N]. However, it is more convenient to formulate our setting in the form of an infinite system, in particularly because we do not want to assume that the channel state ϕ is a product type. This corresponds to the possibility that our quantum source has a memory effect. It is the main point in our stationary source coding theorem that the mean entropy of the channel state appears in the role of source coding rate. First we present our positive source coding theorem for a completely ergodic source. The result says that that the source coding rate may approach the mean entropy while we can keep the fidelity arbitarily good. Theorem 3.1 Let H be a finite dimensional Hilbert space, d := dim H, and ϕ be a completely ergodic state on B(H) with mean entropy s. Then for every ε, δ > 0 there exists n ε,δ N such that for n n ε,δ there is a subspace K n (ε, δ) of H n such that (i) log dim K n (ε, δ) <n(s + δ) and (ii) for every decomposition D n = m i=1 p id (i) one can find an encoding D (i) D (i) with density matrices D (i) supported in K n (ε, δ) such that the fidelity F := m i=1 p itr D D(i) i exceeds 1 ε. The negative part of the coding theorem tells that the source coding rate cannot exceeds the mean entropy when the fidelity is good. Theorem 3.2 Let H be a finite dimensional Hilbert space, d := dim H, and ϕ be a completely ergodic state on B(H) with mean entropy s. Then for every ε, δ > 0 there exists n ε,δ N such that for n n ε,δ (i) for all subspaces K n (ε, δ) of H n with the property log dim K n (ε, δ) <n(s δ) and (ii) for every decomposition D n = m i=1 p id (i) and for every encoding D (i) D (i) with density matrices D (i) supported in K n (ε, δ), the fidelity F := m i=1 p itr D (i) D(i) is smaller than ε. 87

The detailed proofs are given in the paper [15]. This result extends Schumacher s source coding theorem obtained originally for memoryless channels, see [16] and [9]. It is worthwile to note that [8] contains more material on quantum coding. References [1] Araki, H. (1977): Relative entropy of states of von Neumann algebras II, Publ. RIMS Kyoto Univ. 13, 173 192 [2] Araki, H. (1987): Recent progress on entropy and relative entropy, in VIIIth Int. Cong. on Math. Phys. (World Sci. Publishing, Singapore) 354 365 [3] Csiszár, I. (1975): I-divergence geometry of probability distributions and minimization problems, Ann. Prob. 3, 146 158 [4] Donald, M.J. (1990): Relative Hamiltonian which are not bounded from above, J. Functional Anal. 91, pp. 143 173 [5] A. Fujiwara, H. Nagaoka (1994): Capacity of memoryless quantum communication channels, Math. Eng. Tech. Rep. 94-22, University of Tokyo [6] Hiai, F., Ohya, M. Tsukada, M. (1981): Sufficiency, KMS condition and relative entropy in von Neumann algebras, Pacific J. Math. 96, 99 109 [7] Hiai, F., Petz, D. (1991): The proper formula for relative entropy and its asymptotics in quantum probability, Comm. Math. Phys. 143, 99 114 [8] A.S. Holevo (1998): Quantum coding theorems, Russian Math. Surveys, 53, 1295 1331 [9] R. Jozsa, B. Schumacher, A new proof of the quantum noiseless coding theorem, J. Modern Optics, 1994. [10] Kosaki, H. (1986): Relative entropy for states: a variational expression, J. Operator Theory 16, 335 348 [11] Ohya, M (1983): On compound state and mutual information in quantum information theory, IEEE Trans. Information Theory, 29, 770 777 [12] Ohya, M., Petz, D. (1993): Quantum Entropy and Its Use (Springer-Verlag, Berlin, Heidelberg, New York) [13] M. Ohya, D. Petz, N. Watanabe (1997): On capacities of quantum channels, Prob. Math. Stat. 17, 179 196 [14] Petz, D. (1992): Characterization of the relative entropy of states of matrix algebras, Acta Math. Hungar. 59, 449 455 [15] Petz, D., Mosonyi, M. (1999): Stationary quantum source coding, to be published [16] B. Schumacher (1995): Quantum coding, Phys. Rev. A 52, 2738 2747 [17] Uhlmann, A. (1977): Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in an interpolation theory, Commun. Math. Phys. 54, 21 32 [18] Umegaki, H. (1962): Conditional expectations in an operator algebra IV (entropy and information),kodaimath.sem.rep.14, 59 85 88

Adaptive Quantum Measurements (Summary) H.M. Wiseman School of Science, Griffith University, Nathan 4111, Brisbane, Queensland, AUSTRALIA,-J\ / GU -^._/ v In this presentation I will: I Review the general theory of quantum measurements; II Introduce incomplete and adaptive measurements; III Describe different formulations of conditioned state evolution; IV Discuss unraveling the master equation; V Present the theory of dyne measurements; VI Derive the theory of complete dyne measurements; VII Apply this to adaptive phase measurements; and VIII Conclude. I. QUANTUM MEASUREMENT THEORY A. Orthodox Quantum Measurement Theory An observable Λ is represented by an operator where Λ= λ λ R, Π λ Π λ = δ λ,λ Π λ, λπ λ, (1.1) Π λ =1. (1.2) The probability for obtaining the result λ is where λ P λ(t) =Tr[ ρ λ (t + T )], (1.3) ρ λ (t + T )=Π λ ρ(t)π λ. (1.4) Here ρ(t) is the system state matrix at time t (the start of the measurement). The system state matrix at the end of the measurement (time t + T )is ρ λ (t + T )= ρ λ (t + T )/P λ. (1.5) If the initial state is pure then we can write ψ λ (t + T ) =Π λ ψ(t). (1.6) B. More General Quantum Measurements Still restricting the discussion to efficient measurements (in which final state is pure if initial state is pure), we can generalize orthodox measurements: Λ trash, (1.7) Π λ Ω λ. (1.8) The {Ω λ (T )} are arbitrary, subject only to Ω λ Ω λ =1. (1.9) λ I will call them measurement operators. The unnormalized conditioned state is ψ λ (t + T ) =Ω λ ψ(t). (1.10) The probability for result λ is P λ (t) = ψ λ (t + T ) ψ λ (t + T ). (1.11) C. Most General Quantum Measurements We now allow for inefficient measurements (in which the final state may be mixed even if initial state is pure). The unnormalized conditioned state is ρ λ (t + T )= k We can also write this as Ω λk ρ(t)ω λk. (1.12) ρ λ (t + T )=O λ ρ(t), (1.13) where the operation [1] O λ = k J [Ω λk], where J [A]B ABA. (1.14) The probability for the result λ is P λ (t) =Tr[ ρ λ (t + T )] = Tr[ρ(t)F λ ], (1.15) where the effect [1] F λ = k Ω λk Ω λk. The only restriction on {O λ } is conservation of probability which requires F λ =1. (1.16) λ 89

II. INCOMPLETE AND ADAPTIVE MEASUREMENTS A. Classifying Measurements There are many different, overlapping classes based on the properties of {O λ }. For example Back-action evading (and efficient) [2]: Sharp: λ rank [F λ ]=1[3] i.e. λ O λ = k λ Ω λ =Ω λ. (2.1) J [ θ λk φ λ ]. (2.2) C. Adaptive Measurements If a measurement is incomplete, ρ λ (t + T ) depends on ρ(t) and on λ. Subsequent measurements can reveal more information about the preparation of ρ(t). To optimize the information gained, subsequent measurements should (in general) depend on λ. That is, the optimum scheme is adaptive. I will consider the example of adaptive phase measurements. To understand this we first need to understand Linear, nonlinear and semilinear formulations of conditioned state evolution Lindblad master equations Dyne measurements ( θ and φ denote unnormalized states.) Here I introduce a (perhaps new) class. Complete: III. FORMULATIONS FOR THE CONDITIONED STATE EVOLUTION See for example Ref. [4]. λ O λ = jk J [ θ λk φ λj ]. (2.3) A. Nonlinear Obviously sharp = complete, but not vice versa except for efficient measurements. Generate ρ λ (t + T )=O λ ρ(t)/p λ (t) (3.1) B. Why Complete? Because the final state is determined solely by the measurement result ρ λ (t + T )= O λρ(t) (2.4) P λ (t) jk = θ λk φ λj ρ(t) φ λj θ λk jk θ (2.5) λk θ λk φ λj ρ(t) φ λj jk = θ λk θ λk jk θ (2.6) λk θ λk so no more information about ρ(t) can be obtained from measuring the system again. For efficient measurements, complete sharp: λ Ω λ = θ λ φ λ, (2.7) and an incomplete measurement is defined by λ :Ω λ θ λ φ λ. (2.8) An example of an incomplete measurement on a harmonic oscillator [a, a ]=1is Ω 1 = a(aa ) 1/2 sin(ɛaa ), Ω 2 =cos(ɛa a) (2.9) according to the probability distribution and weight all equally. Generate at random, and weight by Generate P λ (t) =Tr[O λ ρ(t)] (3.2) B. Linear ρ λ (t + T )=O λ ρ(t) (3.3) Tr[ ρ λ (t + T )] = P λ (t). (3.4) C. Semilinear ρ λ (t + T )=O λ ρ(t)/q λ (t) (3.5) according to an arbitrary probability distribution Q λ (t), and weight by Tr[ ρ λ (t + T )] = P λ (t)/q λ (t). (3.6) 90

IV. UNRAVELING THE MASTER EQUATION V. DYNE DETECTION A. Unconditioned state evolution If we ignore measurement results then we get the unconditioned state matrix ρ(t + T )= ρ λ (t + T )P λ (t) (4.1) λ = ρ λ (t + T ) (4.2) λ = ρ λ (t + T )Q λ (t) (4.3) λ = O unc ρ(t) (4.4) where the unconditioned evolution operation is O unc = λ O λ. (4.5) For T = dt, O unc =1+O(dt). The most general form in this case is the Lindblad master equation [5] O unc =1+Ldt, (4.6) Instead of directly detecting the light from system, one can interfere it with coherent light (the local oscillator ) before detection. For a strong local oscillator the photon flux is too great for individual photons to be resolved. Instead, one measures a photocurrent I t R. If we discretize time in intervals [t, t + dt), then we can normalize the photocurrent I(t) (with dimension time 1/2 )sothat I 2 t =1/dt. (5.1) A. Dyne Detection Measurement Operators For efficient detection, we must use measurement operators indexed by the continuous variable I t [7] Ω(I t )= Q (I t ) [ 1+dt ( I t e iφ γa 1 2 γa a )], (5.2) where Φ is the local oscillator phase and Q (I t )=(dt/2π) 1/2 exp ( ) 1 2 dti2 t. (5.3) Lρ = i[h, ρ]+ µ D[c µ ]ρ, (4.7) The Q-distribution for I t is that appropriate for Gaussian white noise dw/dt: where D[c]ρ cρc 1 2 c cρ 1 2 ρc c ; H = H (4.8) I t Q di t Q(I t ) I t =0 (5.4) (It ) 2 Q di t Q(I t )(I t ) 2 =1/dt. (5.5) From these moments, we can show that B. Unraveling the master equation There are many different measurements {O λ } giving the same unconditioned evolution O unc. For O unc =1+Ldt these are called different unravelings of the master equation (ME) [6]. Each unraveling describes an ensemble of quantum trajectories for the system state which on average reproduces the ME. The different unravelings correspond to different detection schemes. For example, efficient direct detection of photons escaping a single-mode cavity gives quantum jumps (for result 1): Ω 1 = γdta; Ω 0 =1 1 2 γdta a, (4.9) where γ is the cavity linewidth. The ME is ρ = Lρ = γd[a]ρ. (4.10) di t Ω (I t )Ω(I t )=1, (5.6) di t J [Ω(I t )] = 1 + Ldt. (5.7) B. Linear Quantum Trajectories Measurement operators for dyne detection are ideal for semilinear evolution formulation. ψ I (t + dt) = Ω(I t) ψ(t) (5.8) Q(It ) = [ 1+dt ( I t e iφ γa 1 2 γa a )] ψ(t) Thus ψ I (t) obeys a linear stochastic Schrödinger equation [7] d ψ I (t) = dt ( I t e iφ γa 1 2 γa a ) ψ I (t). (5.9) 91

Here the I subscript shows the state is conditioned on the complete record I [0,t) = {I u : u [0,t)}. The results I t are generated according to the ostensible [4] distribution Q(I t ), so ostensibly d ψ I (t) = ( γdw(t)e iφ a 1 2 γdta a ) ψ I (t). (5.10) The actual probability for the record I [0,t) is P act ( I[0,t) ) = Post ( I[0,t) ) ψi (t) ψ I (t). (5.11) where the effect is F (A, B) =P ost (A, B) φ A,B φ A,B, (6.4) where P ost (A, B) isthatimpliedby P ost (I t )=Q(I t ) t. (6.5) i.e. P ost (A, B) is that which would result if I(t) =dw (t)/dt. (6.6) C. Solving the Linear Quantum Trajectory Let γ =1so ρ = Lρ = D[a]ρ and d ψ I (t) = ( dw (t)e iφ a 1 2 dt a a ) ψ I (t). (5.12) Now using the Itô calculus, ψ I (t + dt) =exp ( 1 2 a adt ) ( ) exp dw (t)e iφ(t) a ( ) exp 1 2 e 2iΦ(t) a 2 dt ψ I (t + dt). This suggests the following solution [4] which can be verified by substitution where ψ I (t) =exp ( 1 2 a at ) exp ( 1 2 S t a2 + Rt a) ψ(0), (5.13) R t = t S t = 0 t e iφ(s) e s/2 I(s)ds (5.14) 0 e 2iΦ(s) e s ds. (5.15) That is to say, the solution at time t depends not on the full function I [0,t), but only on these two complex functionals. 1 VI. COMPLETE DYNE MEASUREMENTS A dyne measurement is only complete for t : ψ I ( ) = 0 0 exp ( 1 2 B a 2 + A a ) ψ(0) (6.1) 0 φ A,B ψ(0) (6.2) where A = R,B = S. A and B are sufficient statistics for I [0,t),and P act (A, B) =Tr[ρ(0)F (A, B)], (6.3) A. Example: Homodyne Homodyne means the local oscillator has the same frequency as the system, so Φ is constant and A ost = e iφ e t/2 dw (t) e iφ X Φ. (6.7) 0 B = e 2iΦ e t dt e 2iΦ (6.8) and These imply which gives 0 P ost (A, B) d 2 Ad 2 B e X2 Φ /2 2π dx Φ. (6.9) (ae iφ + a e iφ ) φ A,B = X Φ φ A,B, (6.10) F (A, B) F hom (X Φ )= X Φ X Φ. (6.11) Thus a completed homodyne measurement is a measurement of one quadrature of the field. B. Example: Heterodyne Heterodyne means the local oscillator has a different frequency so Φ = t with 1and A ost = B = These imply 0 0 e i t t/2 dw (t) α (6.12) e 2it t dt 0. (6.13) P ost (A, B) d 2 Ad 2 B e α 2 π d 2 α (6.14) 1 Φ(s) may depend on I [0,s). 92

and φ A,B = e α 2 /2 α (6.15) which gives F (A, B) F het (α) =π 1 α α. (6.16) Thus a completed heterodyne measurement is a measurement of the complex amplitude α = re iφ. This is the standard way to measure an optical phase: F het (φ) = 0 rdrπ 1 re iφ re iφ. (6.17) VII. ADAPTIVE PHASE MEASUREMENTS A better way to measure phase is adaptively. Wecan adjust Φ(t) in real time to make a homodyne measurement of the estimated phase quadrature: Φ(t) =ˆφ(t)+π/2. (7.1) Here ˆφ(t) is a phase estimate based on R t,s t. The simplest example uses [8] ˆφ(t) =argr t. (7.2) Now ostensibly so R t = t 0 e iφ(s) e s/2 dw (s), (7.3) dr t = e iφ(t) e t/2 dw (t) =i R t R t e t/2 dw (t). (7.4) This has the solution ( ) dw (t) A = R =exp i, (7.5) 0 et 1 which is random number of unit norm. A. Special Case: at most one photon in field VIII. CONCLUSIONS 1. Generalized measurement theory is needed to describe real measurements e.g. in quantum optics. 2. Measurements may be incomplete which implies that future measurements will provide more information about the state preparation. 3. To optimize the information obtained, we can use adaptive control of future measurements. 4. Dyne measurements on optical systems are well described by linear quantum trajectories. 5. Analytical solutions are possible for homodyne, heterodyne, and a simple adaptive scheme. 6. For a system with at most one photon, the adaptive phase measurement is superior to the heterodyne phase measurement. 7. Continuing research: adaptive phase measurements other applications of linear quantum trajectories numerical simulations using nonlinear quantum trajectories generalization of quantum trajectory theory (e.g. to non-markovian systems) If there is at most one photon in the field then F (A, B) can be replaced by F (A) =P ost (A)( 0 + A 1 )( 0 + A 1 ). (7.6) The effect for a measurement of φ =arga is F (φ) = A d A F (A). (7.7) [1] E.B. Davies, Quantum Theory of Open Systems (Academic 0 Press, London, 1976). Since the adaptive technique implies A = 1, [2] V.B. Braginsky and F.Y. Khalili, Quantum Measurement F adapt (φ) = 1 2π φ φ, φ = 0 + eiφ 1, (7.8) (Cambridge University Press, Cambridge, 1992). [3] H. Martens and W.M. de Muynck, Found. Phys. 20, 255 (1990); ibid., 357 (1990) which is the optimum effect for a phase measurement on [4] H.M. Wiseman, Quantum Semiclass. Opt 8, 205 (1996). this system. [5] G. Lindblad, Commun. math. Phys. 48, 199 (1976). By contrast the heterodyne effect gives [6] H.J. Carmichael, An Open Systems Approach to Quantum ( ) Optics (Springer-Verlag, Berlin, 1993). π π 1 F het (φ) = 2 F adapt(φ)+ 1 2 2π. (7.9) [7] P. GoetschandR. Graham, Phys. Rev. A50, 5242 (1994). [8] H.M. Wiseman, Phys. Rev. Lett. 75, 4587 (1995). i.e. as if it were an optimum measurement 88% of time, but gave a random result 12% of time. 93

94 Workshop on Stochastics and Quantum Physics Aarhus 21/10/99 -- 26/10/99 Stochastic Analysis as a tool for the foundations of Quantum Mechanics J.C. Zambrini, GFM, Av. Prof. Gama Pinto 2, 1649-003 LISBOA, PORTUGAL A short story of quantum mechanics and probability. To make this story short indeed, I'll consider only the probabilistic status of pure states, for 2 h systems of Hamiltonian H = - + V. For mixture of pure states, there is nothing special 2 about the probability involved. There are basically two kinds of motivations for the basic introduction of probability theory in Quantum Mechanics (QM): a) Technical Motivations Origin: M. Kac's work (1949) inspired by a lecture of R. Feynman on his path integral representation of solutions of the wave equation in L 2 (IR n ): ψ ih = Hψ t ψ( x,0) = ϕ( x) i S[ ω, t] i.e. ψϕ ( x, t ) = ϕ( ω(0)) eh Dω tx, Ω = { ω C([ 0, t], IR ) s.t ω(t) = x } n where Dωdenotes the flat "measure" d ω( τ) and t 0 τ t S[ ω, t ] = 1 2 ωç( τ) V( ω( t)) d τ S0[ ω, t] V( ω( t)) d τ < is the classical action 0 2 01424 3 functional of the system. t perturbation

95 i S 0[ ω; t] Unfortunately, lim e h d ωτ ( ), where the limit is in L 2 sense when the disc τ {t 1, K, t j= t} discretization goes to zero, is not σ - additive there is no such complex measure on Ω t,x! This is not only a technical detail, as some radical physicists would love to think. It means that there is no such thing as FeynmanÕs fundamental integral over a space of continuous paths in Quantum Theory and that any consequence drawn from it is under legitimate suspicion. η -h * = Hη* On the other hand, after 0 < t it, Schrodinger Ç equation t η* ( x,0) = χ( x) i.e. the heat equation. In this case, it is well known that with coeff 1 S [ ; ] 0 ω t lim e h h dωτ ( ) = d µ w, the Wiener measure disc h and, for a large class of potentials V the following path integral representation makes sense: η * t x χ ( x, t) = t, - 1 Ε χ( w ( 0)) e h V( w( τ)) dτ 0 where Ε t,x denotes the conditional expectation given that w( τ ) = x. NB: The Wiener process w( τ ) is used exclusively as a technical tool here. Its familiar irreversible properties have nothing to do with the ( reversible) one of free quantum dynamics. The way probability enters here has nothing to do with the way it enters in Quantum Mechanics according to Born: ψψ ( xtdx, ) = Prob {system B at time t }. All crucial features of quantum mechanical probability ( interference etcé) are missing. B There are many variations on KacÕs theme since 1949. For example, one can introduce complex Markov processes with diffusions coefficient ( 1 + i) h the way Doss did (1980). Then KacÕs representation holds for Schršdinger equation but for a narrow class of potentials V and of solutions ψ t. Of course, Born interpretation of ψ is lost. Unfortunately, the structure of Quantum theory lies more in this interpretation than in an underlying hypothetical process.

96 b) Foundations of Q.M. b 1 ) SchršdingerÕs (1932) Idea: To construct a model of classical physics (in order to avoid any metaphysical contamination ) where Probability enters like in Q.M. Consider the Cauchy problem for the free diffusion equation: 2 η * 1 η = * 2 = H 0η*, with η* = ρ-t/2 given. We know all about it since t 2 x T / 2 EinsteinÕs time. Now consider the new assumption: the observer knows another 2 η η probability density in the future ρ -T/2 ( dy). For + = this is an O.K data. For 2 t x * { ρ and ρ }, the most probable evolution is ρ ( x) = ηη ( x, td ) x, compatible with these -T/2 T/2 data. This ( Euclidean) Born interpretation lies at the foundation of (EQM) Euclidean T T Quantum Mechanics. On I = 2, 2, the interpolating real valued diffusion Z t is well defined (this is a Bernstein diffusion). Its qualitative properties are very Ònon classicaló. 1/ For example: If η = ( ρ 2 S e )( x, t) is a positive free solution, * * * η ( x l, t) + η ( x + l, t), ( for l= cste) is another one. The probability of the superposed Bernstein diffusion is t 1/2 1/2 ps( x, t) = p( x l, t) + p( x+ l, t) + 2p ( x l, t) p ( x+ l, t)cosh[ S( x l, t) S( x+ l, t)] 14444444444244444444443 Euclidean Interference Observe that if the phase S was purely imaginary, p s would coincide with the quantum interference formula. So, in spite of what we are told all the time, there are classical experiments, indeed, with apparently non-classical probabilistic outcomes. Also notice that the abovementioned decomposition of the positive solution η * involves a term, ρ, even under time-reversal and a phase S odd under this transformation. Clearly, this suggests that our Euclidean counterpart of the complex conjugate is 1 / 2 S 1 / 2 + S η* = ( ρ e )( x, t) η = ( ρ e )( x, t) justifying SchršdingerÕs interpretation of the pair of adjoint heat equations as counterpart of the quantum wave equation and its complex conjugate. Of course, well defined path integral respresentations of η * and η are now available. The physically relevant Bernstein diffusions are Markovian. But, in contrast with most Markovian processes familiar in physics, they respect the time-symmetry of Markov properties, precisely expressed by Ε[( fz) P F ] = Ε[( fz) z,z ], f bounded t s u t s u

97 measurable, where P, F denote, respectively, the past (increasing) and future (decreasing) filtrations. s u b 2 ) Nelson Stochastic Mechanics (1966): Construction of a real valued Markov process 2 compatible with Born interpretation: ψt dx = ρ( x, t) d x, t IR. But a careful analysis shows that NelsonÕs theory is, in fact, a probabilistic interpretation of Bohm theory. Any perturbation of Wiener measure involves not the physical (local) V but Bohm (nonlocal) 1/ 2 ρ potential: ϑ( xt, ) = h 2 ( x, t) V ( x). 1/ 2 ρ This implies, among other disagreements, that all the (well defined) multi times correlations of NelsonÕs processes are without relations with quantum mechanical predictions. b 3 ) Change of the ( Kolmogorovian) foundations of probability theory: Some of the participants in this meeting defend this radical point of view. I intend to summarize recent progress of EQM. Our ultimate goal is to embed regular Quantum Mechanics in a nonprobabilistic framework but much closer to probability theory than Feynman path in tegral approach and, a fortiori, than QM in Hilbert space. The only hope to convince physicists that the detour via stochastic analysis makes sense is to use it for the introduction of new conceptual (and then technical) results in regular QM. So we want to use stochastic analysis to widen the foundations of regular QM through a better analogy than the ones already known, but without claiming that QM is a true (Kolmogorovian) probabilistic theory. Actually, we believe that this claim is hopelessly wrong. A few successful examples of the above mentioned strategy are known today. IÕll sketch their main ideas and results. In particular, it has provided a fresh insight on quantum symmetries, always richer than the one given by the Hilbert space approach. This is in this sense that we are entitled to believe that stochastic analysis is indeed a natural tool for quantum mechanics, the only physical theory where probability lies at the foundations and, we are told, in an irreducible way.

98 References a) For Physicists: J.C.Z. ÒFrom Quantum Physics to Probability Theory and BackÓ in ÒChaos, the interplay between Stochastic and Deterministic behaviouró, Springer Lecture Notes in Physics, N¼ 457, Edit. P. Garbaczewski et al., Spinger, Berlin 1995. b) For Mathematical Physicists: - M. Thieullen, J.C.Z., ÒProbability and Quantum Symmetries IÓ, Ann. Inst. Henri PoincarŽ (Physique ThŽorique), Vol. 67, n¼ 3, (1997) p297. - S. Albeverio, J. Rezende, J.C.Z., ÒProbability and Quantum Symmetries IIÓ, in preparation. c) For Mathematicians: - A.B. Cruzeiro, J.C.Z., ÒMalliavin Calculus and Euclidean Quantum Mechanics I. Functional CalculusÓ, Journal of Funct. Analysis, Vol. 96, N¼ 1, Feb. 15 1991, p62. - A.B. Cruzeiro, Wu Liming, J.C.Z., ÒBernstein processes associated with a Markov ProcessÓ, to appear in Proceedings of the Third International Workshop on Stochastic Analysis and Mathematical PhysicsÓ, R. Rebolledo (Ed.), BirkhŠuser Progress in Probability Series, to appear. - M. Thieullen, J.C.Z., ÒProbability Theory and Related FieldsÓ, 107 (1997), 401.

White noise calculus for fractional Brownian motion and application to stochastic partial differential equations Bernt Øksendal 1),2) 1) Department of Mathematics, University of Oslo Box 1053 Blindern, N-0316, Oslo, Norway, Email: oksendal@math.uio.no 2) Norwegian School of Economics and Business administration, Helleveien 30, N-5035, Bergen-Sandviken, Norway Extended abstract 1 The 1-parameter case Recall that if 0 <H<1 then the fractional Brownian motion with Hurst parameter H is the Gaussian process B H (t); t R with mean E(B H (t)) = 0 and covariance E [B H (t)b H (s)] = 1 { t 2H + s 2H t s 2H} 2 for all s, t R. Here E denotes the expectation with respect to the probability law for B H = B H (t, ω). For simplicity we assume B H (0)=0. If H = 1 2, then B H(t) coincides with the standard Brownian motion B(t). If H> 1 2 then B H (t) has a long range dependence, in the sense that if we put r(n) =cov (B H (1), (B H (n +1) B H (n))) then r(n) =. n=1 AMS 1991 subject classifications. Primary 60H05, 60H10; Secondary 90A09, 90A12. Key words and phrases: Fractional Brownian motions; Fractional white noises; Chaos expansion; Wick calculus; Fractal Black and Scholes market; Arbitrage; Price formula; Replicating portfolio. 99

For any H (0, 1) the process B H (t) is self-similar in the sense that B H (αt) has the same law as α H B H (t) for any α>0. See Mandelbrot and Van Ness [MV] for more information about fractional Brownian motion., 1) has been suggested Because of these properties B H (t) with Hurst parameter H ( 1 2 as a useful tool in many applications, including physics and finance [M]. A major difficulty with these processes is that they are not semimartingales and not Markov processes, so many of the powerful tools from stochastic analysis cannot be used for B H (t). However, we shall see that it is possible to develop a white noise calculus based on B H (t), 1 <H<1. 2 We will use this to study Itô type stochastic differential equations driven by fractional white noise W H (t) = db H(t) dt. Moreover, we will introduce a fractional Malliavin calculus and prove a generalized fractional Clark-Haussmann-Ocone formula. This presentation will follow the paper [HØ] closely. 2 The multiparameter case As in [H1], [H2] we define d-parameter fractional Brownian motion B H (x); x =(x 1,...,x d ) R d with Hurst parameter H =(H 1,...,H d ) (0, 1) d as a Gaussian process on R d with mean and covariance We also assume that E[B H (x)b H (y)] = ( 1 2 )d From now on we will assume that E[B H (x)] = 0 for all x R d (2.1) d ( x i 2H i + y i 2H i x i y i 2H i ) (2.2) i=1 B H (0) = 0 a.s. (2.3) 1 <H 2 i < 1 for i =1,...,d. (2.4) In part 2 we will extend the fractional white noise theory to the multiparameter case and use this theory to study some stochastic partial differential equations, driven by multiparameter fractional white noise W H (x) = d B H (x) x 1... x d ; x R d. This presentation will be based on the paper [HØZ]. For example, we will prove the following result from [HØZ]: Theorem a) For any dimension d there is a unique strong solution U(x) : D (S) H satisfying the fractional Poisson equation U(x) = W H (x) ; x D R d (2.5) U(x) =0 for x D. (2.6) 100

The solution is given by U(x) = D G(x, y)w H (y)dy = D G(x, y)db H (y). (2.7) The solution belongs to C( D) C 2 (D). b) If d d H i < 2, (2.8) i=1 then U(x) L 2 (µ ϕ ) for all x D, so the solution exists as an ordinary stochastic field. Remark Note that if H i = 1 for all i then the condition (2.8) coincides with the known 2 condition d<4. But if H i > 1 for all i we may get L 2 (µ 2 ϕ ) solutions also in higher dimensions. Thus it is easier to obtain L 2 (µ ϕ ) solutions the higher the values of H i are. This seems reasonable from the point of view that the paths of B H (x) get more regular as H increases. References [AØU] K. Aase, B. Øksendal, J. Ubøe: White noise generalizations of the Clark-Ocone theorem with application to mathematical finance. Preprint, MaPhySto, Aarhus, 1998. [B] F.E. Benth: Integrals in the Hida distribution space (S). In T. Lindstrøm, B. Øksendal and A.S. Üstünel (editors): Stochastic Analysis and Related Topics. Gordon and Breach 1993, pp. 89 99. [BG] F.E. Benth and H. Gjessing: A non-linear parabolic equation with noise. A reduction method. Potential Analysis (to appear). [D] A. Dasgupta: Fractional Brownian motion: Its properties and applications to stochastic integration. Ph. D Thesis, Dept of Statistics, University of North Carolina at Chapel Hill, 1997. [DH] W. Dai and C.C. Heyde: Itô formula with respect to fractional Brownian motion and its application. J. Appl. Math. Stoch. Anal. 9 (1996), 439 448. 101

[DHP] T.E. Duncan, Y. Hu and B. Pasik-Duncan: Stochastic calculus for fractional Brownian motion. I. Theory. To appear in SIAM Journal of Control and Optimization. [DK] A. Dasgupta and G. Kallianpur: Arbitrage opportunities for a class of Gladyshev processes. To appear in Appl. Math. & Optimization. [DU] L. Decreusefond and A.S. Üstünel: Stochastic analysis of the fractional Brownian motion. Potential Analysis 10 (1999), 177 214. [GN] G. Gripenberg and I. Norros: On the prediction of fractional Brownian motion. J. Appl. Prob. 33 (1996), 400 410. [H1] Y. Hu: Heat equation with fractional white noise potentials. Preprint 1998. [H2] Y. Hu: A class of stochastic partial differential equations driven by fractional white noise. Preprint 1999. [H3] Y. Hu: Itô-Wiener chaos expansion with exact residual and correlation, variance inequalities. Journal of Theoretical Probability, 10 (1997), 835 848. [HKPS] T. Hida, H.-H. Kuo, J. Potthoff and L. Streit: White Noise Analysis. Kluwer 1993. [HØ] Y. Hu and B. Øksendal: Fractional white noise calculus and applications to finance. Preprint, Dept. of Mathematics, University of Oslo 1999. [HØ1] Y. Hu and B. Øksendal: Wick approximation of quasilinear stochastic differential equations. In H. Körezlioglu, B. Øksendal and A.S. Üstünel (editors): Stochastic Analysis and Related Topics, Vol. 5. Birkhäuser 1996, pp. 203 231. [HØS] Y. Hu, B. Øksendal and A. Sulem: Optimal portfolio in a fractional Black & Scholes market. Preprint, Dept. of Mathematics, University of Oslo, No. 13, August 1999. [HØUZ] H. Holden, B. Øksendal, J. Ubøe and T. Zhang: Stochastic Partial Differential Equations. Birkhäuser 1996. [HØZ] Y. Hu, B. Øksendal and T. Zhang: Stochastic partial differential equations driven by multiparameter fractional white noise. Preprint 1999. [K] H.-H. Kuo: White Noise Distribution Theory. CRC Press 1996. [L] A. Le Breton: Filtering and parameter estimation in a simple linear system driven by a fractional Brownian motion. Stat. Prob. Letters 38 (1998), 263 274. [LC] T. Lundgren and D. Chiang: Solution of a class of singular integral equations. Quart. J. Appl. Math. 24, 303 313. 102

[Li] S.J. Lin: Stochastic analysis of fractional Brownian motion. Stochastics and Stochastic Reports 55 (1995), 121 140. [LØU] T. Lindstrøm, B. Øksendal and J. Ubøe: Wick multiplication and Itô-Skorohod stochastic differential equations. In S. Albeverio, J.E. Fenstad, H. Holden and T. Lindstrøm (editors): Ideas and Methods in Mathematical Analysis, Stochastics, and Applications, Cambridge Univ. Press, 1992, pp. 183 206. [M] B.B. Mandelbrot: Fractals and Scaling in Finance: Discontinuity, Concentration, Risk. Springer-Verlag, 1997. [MV] B.B. Mandelbrot and J.W. Van Ness: Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10 (1968), 422 437. [NVV] I. Norros, E. Valkeila and J. Virtamo: An elementary approach to a Girsanov formula and other analytic results on fractional Brownian motions. Bernoulli 5 (1999), 571 587. [Ø] B. Øksendal: Stochastic Differential Equations. Springer, fifth edition, 1998. [PT] J. Potthoff and M. Timpel: On a dual pair of smooth and generalized random variables. Potential Analysis 4 (1995), 637 654. [R] L.C.G. Rogers: Arbitrage with fractional Brownian motion. Math. Finance 7 (1997), 95 105. [S] A. Shiryaev: On arbitrage and replication for fractal models: In A. Shiryaev and A. Sulem (editors): Workshop on Mathematical Finance, INRIA, Paris 1998. [So] T. Sottinen: Fractional Brownian motion, random walks and binary market models. Preprint, University of Helsinki, June 1999. [Sz] G. Szego: Orthogonal Polynomials. Amer. Math. Soc. Colloq. Pub. 23, Providence, R.I. 1967. [T] S. Thangavelu: Lectures of Hermite and Laguerre Expansions. Princeton University Press, 1993. [V] E. Valkeila: On some properties of geometric fractional Brownian motion, Preprint, University of Helsinki, May 1999. 103

3 Workshop Program (revised) Thursday October 21 (in Auditorium G2, building 532) 09.00-9.50 Registration and coffee/tea 10.10-11.00 11.10-12.00 Chairman: Ole E. Barndorff-Nielsen Bernt Øksendal: Short course on Wick Products, Malliavin Calculus and their applications in Physics. Bernt Øksendal: Short course on Wick Products, Malliavin Calculus and their applications in Physics. 12.30-14.00 Lunch 14.00-14.50 Coffee/tea 15.10-16.00 16.10-17.00 Chairman: Klaus Mølmer Denes Petz: Relative entropy in quantum information theory. Ian Percival: Generalized Bell inequalities. Uffe Haagerup: Spectra of random matrices and random operators on Hilbert spaces. 17.00-18.00 Welcome Reception 104

Friday October 22 (in Auditorium G2, building 532) Chairman: Richard Gill François Bardou: 9.00-9.50 Stochastic wave-functions, Lévy flights and quantum evaporation. Coffee/tea 10.10-11.00 11.10-12.00 Bernt Øksendal: Short course on Wick Products, Malliavin Calculus and their applications in Physics. Bernt Øksendal: Short course on Wick Products, Malliavin Calculus and their applications in Physics. 12.30-14.00 Lunch 14.00-14.50 Chairman: Viacheslav Belavkin Günther Mahler: Statistical measures for the characterization of quantum trajectories. Coffee/tea 15.10-16.00 16.10-17.00 Jean Claude Zambrini: Stochastic analysis as a tool for the foundations of quantum mechanics. Eugene Polzik: Short introduction to Quantum Optics & Lab-Tour. 105

Saturday October 23 (in Auditorium G2, building 532) Chairman: Göran Lindblad Alexander Holevo: 9.00-9.50 Coding Theorems for Quantum Channels. Coffee/tea 10.10-11.00 11.10-12.00 12.00-13.20 Lunch Viacheslav P. Belavkin: Quantum Stochastics as a Boundary Value Problem. Viacheslav P. Belavkin: Classification of Quantum Noise. 13.20-14.10 Coffee/tea 14.30-15.20 15.30-16.20 Chairman: Günther Mahler Alberto Barchielli: Quantum stochastic models of two-level atoms. Hans Maassen: Ergodicity of continuous measurement in quantum optics. Serge Massar: How much classical communication is required to simulate quantum communication and quantum entanglement?. 16.30 Departure for Excursion/Conference Dinner 106

Monday October 25 (in Auditorium G1, building 532) Chairman: François Bardou Asher Peres: 9.00-9.50 Relativistic Quantum Measurement. Coffee/tea 10.10-11.00 11.10-12.00 12.00-12.20 12.30-14.00 Lunch Howard Carmichael: Physical principles of quantum trajectories. Howard Wiseman: Monitoring Open Quantum Systems: Theory and Applications to Adaptive Quantum Measurements. Elena R. Loubenets: The Quantum Stochastic Evolution of an Open System under Continuous in Time Non-demolition Measurement. 14.00-14.50 Coffee/tea 15.10-16.00 16.10-17.00 Chairman: Alexander Holevo Richard Gill: Quantum asymptotic statistics and non-locality without entanglement. Alexander Gottlieb: Quantum molecular chaos and the dynamics of mean-field spin models. Yuri Yu. Lobanov: Stochastic calculations in quantum physics via numerical integration in metric spaces. Tuesday October 26 (in Auditorium G1, building 532) Chairman: Elena Loubenets Inge Helland: 9.00-9.50 Experiments, symmetries and quantum mechanics. Coffee/tea 10.10-11.00 11.10-12.00 12.30-14.00 Lunch Peter Høyer: The Subgroup Problem via State Distinguishability. Göran Lindblad: Gaussian maps and processes in quantum systems. 107

4 List of participants Luigi Accardi Università degli Studi di Roma Tor Vergata Centro Vito Volterra Via di Tor Vergata, snc I-00133 Roma, Italy Email: accardi@volterra.mat.uniroma2.it WWW: http://volterra.mat.uniroma2.it Alberto Barchielli Universita di Milano Dipartimento di Matematica Via Saldini 50 I-20133 Milano, Italy Email: albbar@pop2.mate.polimi.it Francois Bardou Department of Physics University of Newcastle Newcastle Upon Tyne, NE1 7RU United Kingdom Email: bardou@morgane.u-strasbg.fr Ole E. Barndorff-Nielsen Department of Mathematical Sciences University of Aarhus DK-8000 Aarhus C Denmark Email: oebn@imf.au.dk Viacheslav P. Belavkin School of Mathematical Sciences University of Nottingham Nottingham NG7 2RD United Kingdom Email: vpb@maths.nott.ac.uk Fred Espen Benth Department of Mathematics University of Oslo P.B. 1053 Blindern N-0316 Oslo Norway Email: fredb@math.uio.no 108

Howard Carmichael University of Ulm Department of Quantum Physics D-89069 Ulm Germany Email: Howard.Carmichael@physik.uni-ulm.de Richard D. Gill University of Utrecht Mathematical Institute Budapestlaan 6 NL-3585 CD Utrecht, The Netherlands Email: Richard.Gill@math.ruu.nl WWW: http://www.math.uu.nl/people/gill/ Alexander Gottlieb Lawrence Berkeley National Laboratory 1 Cyclotron Road, Mail Stop 50A-1148 94720 Berkeley, CA USA Email: gottlieb@math.berkeley.edu WWW: http://www.lbl.gov/ gottlieb/ Svend Erik Graversen Department of Mathematical Sciences University of Aarhus DK-8000 Aarhus C Denmark Email: matseg@imf.au.dk Inge S. Helland Department of Mathematics University of Oslo P.O.Box 1053 Blindern N-0316 Oslo, Norway Email: ingeh@math.uio.no WWW: http://www.math.uio.no/ ingeh/ Alexander S. Holevo Steklov Mathematical Institute Russian Academy of Sciences Moscow Russia Email: holevo@genesis.mi.ras.ru 109

Peter Høyer BRICS Department of Computer Science University of Aarhus DK-8000 Aarhus C, Denmark Email: hoyer@brics.dk Uffe Haagerup Matematisk Institut Odense Universitet Campusvej 55 DK-5230 Odense M, Denmark Email: haagerup@imada.sdu.dk Jens Ledet Jensen Department of Mathematical Sciences University of Aarhus DK-8000 Aarhus C Denmark Email: jlj@imf.au.dk Peter E. Jupp School of Mathematical and Computational Sciences University of St. Andrews North Haugh St Andrews KY16 9SS, Scotland Email: pej@st-andrews.ac.uk Martin Leitz-Martini University of Tübingen Mathematical Institute Auf der Morgenstelle 10 D-72076 Tübingen, Germany Email: martin.leitz@uni-tuebingen.de Mykola Leonenko Department of Mathematics Kiev University Volodimirska str. 64 252601 Kyiv Ukraine Email: leonenko@aki.pp.kiev.ua 110

Göran Lindblad Theoretical Physics Royal Institute of Technology SE-100 44 Stockholm Sweden Email: gli@theophys.kth.se Yuri Yu. Lobanov Lab. of Computing Techniques and Automation Joint Institute for Nuclear Research Dubna, Moscow Region 141980 Russia Email: lobanov@main1.jinr.ru Elena R. Loubenets Applied Mathematics Department Moscow State Institute of Electronics and Mathematics Moscow, Russia Email: erl@erl.msk.ru Alessandra Luati Dipartimento di Scienze Statistiche Paolo Fortunati via Belle Arti 40 I-40126 Bologna, Italy Email: luati@stat.unibo.it Hans Maassen University of Nijmegen Toernooiveld 1 6525 ED Nijmegen The Netherlands Email: maassen@sci.kun.nl Günther Mahler Institute for Theoretical Physics I Pfaffenwaldring 57 D-70550 Stuttgart Germany Email: mahler@theo.physik.uni-stuttgart.de 111

David Márquez-Carreras Facultat de Matemátiques Universitat de Barcelona Gran Via 585 E-08007 Barcelona, Spain Email: marquez@mat.ub.es Serge Massar Service de Physique Theorique Universite Libre de Bruxelles CP 225, Bvd du Triomphe B1050 Bruxelles, Belgium Email: smassar@ulb.ac.be Milan Mosonyi Panorama u.90 2083 Solymar Hungary Email: mmm@ludens.elte.hu Klaus Mølmer Institute of Physics and Astronomy University of Aarhus DK-8000 Aarhus C Denmark Email: moelmer@ifa.au.dk Ian Percival Department of Physics Queen Mary and Westfield College Mile End Road London E1 4NS, United Kingdom Email: I.C.Percival@qmw.ac.uk Asher Peres Department of Physics Technion 32000 Haifa Israel Email: peres@photon.technion.ac.il 112

Denes Petz Department of Analysis Mathematics Institute Technical University of Budapest H-1521 Budapest XI, Hungary Email: petz@math.bme.hu Goran Peskir Department of Mathematical Sciences University of Aarhus DK-8000 Aarhus C Denmark Email: goran@imf.au.dk Eugene Polzik Institute of Physics and Astronomy University of Aarhus DK-8000 Aarhus C Denmark Email: polzik@dfi.aau.dk Daniel Terno Department of Physics Technion 32000 Haifa Israel Email: terno@physics.technion.ac.il Steen Thorbjørnsen Dept. of Mathematics Odense University Campusvej 55 DK-5230 Odense M, Denmark Email: steenth@imada.sdu.dk 113

Flemming Topsøe Department of Mathematical Sciences University of Copenhagen Universitetsparken 5 2100 København Ø, Denmark Email: topsoe@math.ku.dk Howard Wiseman School of Science Griffith University Nathan 4111, Brisbane Queensland, Australia Email: H.Wiseman@mailbox.gu.edu.au Jean Claude Zambrini Grupo de Fisica-Matematica Universidade de Lisboa Av. Prof. Gama Pinto, 2 1649-003 Lisboa, Portugal Email: zambrini@alf1.cii.fc.ul.pt Eberhard Zeidler Max Planck Institute for Mathematics in the Sciences Inselstrasse 22 D-04103 Leipzig, Germany Email: ezeidler@mis.mpg.de Bernt Øksendal Department of Mathematics University of Oslo B.P. 1053 Blindern N-0316 Oslo, Norway Email: oksendal@math.uio.no 114