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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].
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
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.
Interactions in noncommutative dynamics
 Comm. Math. Phys
, 2000
"... Abstract. For a fixed C ∗algebra A, we consider all noncommutative dynamical systems that can be generated by A. More precisely, an Adynamical system is a triple (i, B, α) where α is a ∗endomorphism of a C ∗algebra B, and i: A ⊆ B is the inclusion of A as a C ∗subalgebra with the property that ..."
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Cited by 4 (4 self)
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Abstract. For a fixed C ∗algebra A, we consider all noncommutative dynamical systems that can be generated by A. More precisely, an Adynamical system is a triple (i, B, α) where α is a ∗endomorphism of a C ∗algebra B, and i: A ⊆ B is the inclusion of A as a C ∗subalgebra with the property that B is generated by A ∪α(A) ∪α 2 (A) ∪ · · ·. There is a natural hierarchy in the class of Adynamical systems, and there is a universal one that dominates all others, denoted (i, PA,α). We establish certain properties of (i, PA,α) and give applications to some concrete issues of noncommutative dynamics. For example, we show that every contractive completely positive linear map ϕ: A → A gives rise to to a unique Adynamical system (i, B, α) that is “minimal ” with respect to ϕ, and we show that its C ∗algebra B can be embedded in the multiplier algebra of A ⊗ K. 1. Generators The flow of time in quantum theory is represented by a oneparameter group of ∗automorphisms {αt: t ∈ R} of a C ∗algebra B. There is often a C ∗subalgebra A ⊆ B that can be singled out from physical considerations which, together with its time translates, generates B. For example, in nonrelativistic quantum mechanics the flow of time is represented by a oneparameter group of automorphisms of B(H), and the set of all bounded continuous functions of the configuration observables at time 0 is a commutative C ∗algebra A. The set of all time translates αt(A) of A generates an irreducible C∗subalgebra B of B(H). In particular, for different times t1 ̸ = t2, the C∗algebras αt1 (A) and αt2 (A) do not commute with each other. Indeed, no nontrivial relations appear to exist between αt1 (A) and αt2 (A) when t1 ̸ = t2. In this paper we look closely at this phenomenon, in a simpler but analogous setting. Let A be a C∗algebra, fixed throught. Definition 1.1. An Adynamical system is a triple (i,B,α) consisting of a ∗endomorphism α acting on a C ∗algebra B and an injective ∗homomorphism i: A → B, such that B is generated by i(A) ∪ α(i(A)) ∪ α 2 (i(A)) ∪ · · ·. We lighten notation by identifying A with its image i(A) in B, thereby replacing i with the inclusion map i: A ⊆ B. Thus, an Adynamical system is a dynamical system (B,α) that contains A as a C ∗subalgebra in
Characteristic Functions for Ergodic Tuples
, 2006
"... Abstract. Motivated by a result on weak Markov dilations, we define a notion of characteristic function for ergodic and coisometric row contractions with a onedimensional invariant subspace for the adjoints. This extends a definition given by G.Popescu. We prove that our characteristic function is ..."
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Cited by 1 (1 self)
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Abstract. Motivated by a result on weak Markov dilations, we define a notion of characteristic function for ergodic and coisometric row contractions with a onedimensional invariant subspace for the adjoints. This extends a definition given by G.Popescu. We prove that our characteristic function is a complete unitary invariant for such tuples and show how it can be computed.
INTERPOLATIONS BETWEEN BOSONIC AND FERMIONIC RELATIONS GIVEN BY GENERALIZED BROWNIAN MOTIONS
, 1994
"... Please send all correspondence to: ..."
NONCOMMUTATIVE FLOWS I: DYNAMICAL INVARIANTS
, 1995
"... Abstract. We show that a noncommutative dynamical system of the type that occurs in quantum theory can often be associated with a dynamical principle; that is, an infinitesimal structure that completely determines the dynamics. The nature of these dynamical principles is similar to that of the secon ..."
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Abstract. We show that a noncommutative dynamical system of the type that occurs in quantum theory can often be associated with a dynamical principle; that is, an infinitesimal structure that completely determines the dynamics. The nature of these dynamical principles is similar to that of the second order differential equations of classical mechanics, in that one can locate a space of momentum operators, a “Riemannian metric”, and a potential. These structures are classified in terms of geometric objects which, in the simplest cases, occur in finite dimensional matrix algebras. As a consequence, we obtain a new classification of E0semigroups acting on type I factors. Contents