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103
qGaussian processes: Noncommutative and classical aspects
 Commun. Math. Phys
, 1997
"... Abstract. We examine, for −1 < q < 1, qGaussian processes, i.e. families of operators (noncommutative random variables) Xt = at + a ∗ t – where the at fulfill the qcommutation relations asa ∗ t − qa ∗ t as = c(s, t) · 1 for some covariance function c(·, ·) – equipped with the vacuum expectation ..."
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Cited by 64 (2 self)
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Abstract. We examine, for −1 < q < 1, qGaussian processes, i.e. families of operators (noncommutative random variables) Xt = at + a ∗ t – where the at fulfill the qcommutation 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 qanalogue of the Gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on qGaussian 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 qGaussian processes possess a noncommutative kind of Markov property, which ensures that there exist classical versions of these noncommutative processes. This answers an old question of Frisch and Bourret [FB].
Pautrat: “From repeated to continuous quantum interactions”, Annales Henri Poincaré (Physique Théorique
"... We consider the general physical situation of a quantum system H0 interacting with a chain of exterior systems ⊗IN ∗H, one after the other, during a small interval of time h and following some Hamiltonian H on H0 ⊗ H. We discuss the passage to the limit to continuous interactions (h → 0) in a setup ..."
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Cited by 36 (3 self)
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We consider the general physical situation of a quantum system H0 interacting with a chain of exterior systems ⊗IN ∗H, one after the other, during a small interval of time h and following some Hamiltonian H on H0 ⊗ H. We discuss the passage to the limit to continuous interactions (h → 0) in a setup which allows to compute the limit of this Hamiltonian evolution in a single state space: a continuous field of exterior systems ⊗ IR +H. Surprisingly, the passage to the limit shows the necessity for 3 different time scales in H. The limit evolution equation is shown to spontaneously produce quantum noises terms: we obtain a quantum Langevin equation as limit of the Hamiltonian evolution. For the very first time, these quantum Langevin equations are obtained as the effective limit from repeated to continuous interactions and not only as a model. These results justify the usual quantum Langevin equations considered in continual quantum measurement or in quantum optics. We show that the three time scales correspond to the normal regime, the weak coupling limit and the low density limit. Our approach allows to consider these two physical limits altogether for the first time. Their combination produces an effective Hamiltonian on the small system, which had never been described before. We apply these results to give an Hamiltonian description of the von Neumann measurement. We also consider the approximation of continuous time quantum master equations by discrete time ones. In particular we show how any Lindblad generator is obtained as the limit of completely positive maps.
Stochastic Calculus With Respect To Free Brownian Motion And Analysis On Wigner Space
, 1998
"... . We define stochastic integrals with respect to free Brownian motion, and show that they satisfy BurkholderGundy 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 ..."
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Cited by 35 (3 self)
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. We define stochastic integrals with respect to free Brownian motion, and show that they satisfy BurkholderGundy 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 State Reduction
, 2004
"... 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 th ..."
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Cited by 32 (2 self)
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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.
Stochastic Schrödinger equations
 J. Phys. A: Math. Gen
"... 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 es ..."
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Cited by 27 (6 self)
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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
An introduction to quantum filtering
, 2006
"... 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 ..."
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Cited by 24 (13 self)
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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
Chaotic states and Stochastic Integration in Quantum Systems
 Russian Math. Survey
, 1992
"... 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 monoidvalued fields on an atomless ‘spacetime’ ..."
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Cited by 14 (12 self)
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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 monoidvalued fields on an atomless ‘spacetime’ set. A canonical decomposition of the logarithmic conditionally posivedefinite generating functional is constructed in a pseudoEuclidean space, given by a quadruple defining the monoid triangular operator representation and a cyclic zero pseudonorm state in this space. It is shown that the exponential representation in the corresponding pseudoFock space yields the infinitelydivisible generating functional with respect to the exponential state vector, and its compression to the Fock space defines the cyclic infinitlydivisible 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
Fluctuations of quantum currents and unravelings of master equations
, 2007
"... The very notion of a current fluctuation is problematic in the quantum context. We study that problem in the context of nonequilibrium statistical mechanics, both in a microscopic setup and in a Markovian model. Our answer is based on a rigorous result that relates the weak coupling limit of fluctua ..."
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Cited by 11 (3 self)
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The very notion of a current fluctuation is problematic in the quantum context. We study that problem in the context of nonequilibrium statistical mechanics, both in a microscopic setup and in a Markovian model. Our answer is based on a rigorous result that relates the weak coupling limit of fluctuations of reservoir observables under a global unitary evolution with the statistics of the socalled quantum trajectories. These quantum trajectories are frequently considered in the context of quantum optics, but they remain useful for more general nonequilibrium systems. In contrast with the approaches found in the literature, we do not assume that the system is continuously monitored. Instead, our starting point is a relatively realistic unitary dynamics of the full system.
A Discrete Invitation to Quantum Filtering and Feedback Control
, 2009
"... 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 nov ..."
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Cited by 10 (2 self)
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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 finitedimensional 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.