Abstract not found

Abstract. We examine, for −1 < q < 1, q-Gaussian processes, i.e. families of operators (non-commutative random variables) Xt = at + a ∗ t – where the at fulfill the q-commutation relations asa ∗ t − qa ∗ t as = c(s, t) · 1 for some covariance function c(·, ·) – equipped with the vacuum expectation state. We show that there is a q-analogue of the Gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on q-Gaussian processes into corresponding (and much simpler) questions in the underlying Hilbert space. In particular, we use this idea to show that a large class of q-Gaussian processes possess a non-commutative kind of Markov property, which ensures that there exist classical versions of these non-commutative processes. This answers an old question of Frisch and Bourret [FB].

. We define stochastic integrals with respect to free Brownian motion, and show that they satisfy Burkholder-Gundy type inequalities in operator norm. We prove also a version of Ito's predictable representation theorem, as well as product form and functional form of Ito's formula. Finally we develop stochastic analysis on the free Fock space, in analogy with stochastic analysis on the Wiener space. Introduction In this paper we develop a stochastic integration theory with respect to the free Brownian motion. A strong motivation for undertaking this work was provided by two sources. On one hand the stochastic quantization approach to Master Fields, as described in [D], requires the development of a stochastic calculus with respect to free Brownian motion, in order to be implemented in a mathematically rigourous way. On the other hand, the theory of free entropy developped by D. Voiculescu suggests the study of "free" Gibbs states, whose definition is analogous to the classical Gibbs st...

Feedback control of quantum mechanical systems must take into account the probabilistic nature of quantum measurement. We formulate quantum feedback control as a problem of stochastic nonlinear control by considering separately a quantum filtering problem and a state feedback control problem for the filter. We explore the use of stochastic Lyapunov techniques for the design of feedback controllers for quantum spin systems and demonstrate the possibility of stabilizing one outcome of a quantum measurement with unit probability.

A derivation of stochastic Schrödinger equations is given using quantum filtering theory. We study an open system in contact with its environment, the electromagnetic field. Continuous observation of the field yields information on the system: it is possible to keep track in real time of the best estimate of the system’s quantum state given the observations made. This estimate satisfies a stochastic Schrödinger equation, which can be derived from the quantum stochastic differential equation for the interaction picture evolution of system and field together. Throughout the paper we focus on the basic example of resonance fluorescence. 1

Abstract. This paper provides an introduction to quantum filtering theory. An introduction to quantum probability theory is given, focusing on the spectral theorem and the conditional expectation

Abstract. Quantum chaotic states over a noncommutative monoid, a unitalization of a noncommutative Ito algebra parametrizing a quantum stochastic Levy process, are described in terms of their infinitely divisible generating functionals over the simple monoid-valued fields on an atomless ‘space-time’ set. A canonical decomposition of the logarithmic conditionally posive-definite generating functional is constructed in a pseudo-Euclidean space, given by a quadruple defining the monoid triangular operator representation and a cyclic zero pseudo-norm state in this space. It is shown that the exponential representation in the corresponding pseudo-Fock space yields the infinitely-divisible generating functional with respect to the exponential state vector, and its compression to the Fock space defines the cyclic infinitly-divisible representation associated with the Fock vacuum state. The structure of states on an arbitrary Itô algebra is studied with two canonical examples of quantum Wiener and Poisson states. A generalized quantum stochastic nonadapted multiple integral is explicitly

Let g and p denote the Gaussian and Poisson measures on R, respectively. We show that there exists a unique measure e g on C such that under the Segal-Bargmann transform Sg the space L (R; g ) is isomorphic to the space HL (C ; e g ) of analytic L -functions on C with respect to e g . We also introduce the Segal-Bargmann transform Sp for the Poisson measure p and prove the corresponding result. As a consequence, when g and p have the same variance, L (R; g ) and L (R; p ) are isomorphic to the same space HL (C ; e g ) under the Sg and Sp -transforms, respectively. However, we show that the multiplication operators by x on L (R; p ) and on L (R; g ) act quite dierently on HL (C ; e g ). 1.

The engineering and control of devices at the quantum mechanical level—such as those consisting of small numbers of atoms and photons—is a delicate business. The fundamental uncertainty that is inherently present at this scale manifests itself in the unavoidable presence of noise, making this a novel field of application for stochastic estimation and control theory. In this expository paper we demonstrate estimation and feedback control of quantum mechanical systems in what is essentially a noncommutative version of the binomial model that is popular in mathematical finance. The model is extremely rich and allows a full development of the theory while remaining completely within the setting of finite-dimensional Hilbert spaces (thus avoiding the technical complications of the continuous theory). We introduce discretized models of an atom in interaction with the electromagnetic field, obtain filtering equations for photon counting and homodyne detection, and solve a stochastic control problem using dynamic programming and Lyapunov function methods.

1.1 Three approaches to continual measurements................... 3 1.2 Quantum stochastic calculus and quantum optics................ 3 1.3 Some notations: operator spaces............................... 4 2 Unitary evolution and states.............................. 5