Results 1  10
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14
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 expec ..."
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Cited by 120 (7 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].
An introduction to quantum filtering
, 2006
"... 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 as a ..."
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Cited by 79 (19 self)
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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 as a
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 47 (8 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.
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 26 (4 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.
DILATION OF MARKOVIAN COCYCLES ON A Von Neumann Algebra
 PACIFIC JOURNAL OF MATHEMATICS
, 2003
"... We consider normal Markovian cocycles on a von Neumann algebra which are adapted to a Fockfiltration. Every such cocycle k which is Markovregular and consists of completely positive contractions is realised as a conditioned ∗homomorphic cocycle. This amounts to a stochastic generalisation of a rec ..."
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Cited by 15 (7 self)
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We consider normal Markovian cocycles on a von Neumann algebra which are adapted to a Fockfiltration. Every such cocycle k which is Markovregular and consists of completely positive contractions is realised as a conditioned ∗homomorphic cocycle. This amounts to a stochastic generalisation of a recent dilation result for normcontinuous normal completely positive contraction semigroups. To achieve this stochastic dilation we use the fact that k is governed by a quantum stochastic differential equation whose coefficient matrix has a specific structure, and extend a technique for obtaining stochastic flow generators from Markov semigroup generators, to the context of cocycles. Number/exchangefree dilatability is seen to be related to locality in the case where the cocycle is a Markovian semigroup. In the same spirit unitary dilations of Markovregular contraction cocycles on
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 8 (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 4 (4 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.
Four lectures on noncommutative dynamics, in
 Muhly (Eds.), Advances in Quantum Dynamics (Mount
, 2003
"... Abstract. These lectures concern basic aspects of the theory of semigroups of endomorphisms of type I factors that relate to causal dynamics, dilation theory, and the problem of classifying E0semigroups up to cocycle conjugacy. We give only a few proofs here; full details can be found in the author ..."
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Cited by 2 (1 self)
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Abstract. These lectures concern basic aspects of the theory of semigroups of endomorphisms of type I factors that relate to causal dynamics, dilation theory, and the problem of classifying E0semigroups up to cocycle conjugacy. We give only a few proofs here; full details can be found in the author’s upcoming monograph Noncommutative Dynamics and Esemigroups, to be published in
ON THE EXISTENCE OF E0SEMIGROUPS
, 2006
"... Abstract. Product systems are the classifying structures for semigroups of endomorphisms of B(H), in that two E0semigroups are cocycle conjugate iff their product systems are isomorphic. Thus it is important to know that every abstract product system is associated with an E0semigrouop. This was fi ..."
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Abstract. Product systems are the classifying structures for semigroups of endomorphisms of B(H), in that two E0semigroups are cocycle conjugate iff their product systems are isomorphic. Thus it is important to know that every abstract product system is associated with an E0semigrouop. This was first proved more than fifteen years ago by rather indirect methods. Recently, Skeide has given a more direct proof. In this note we give yet another proof by an elementary construction. 1. Formulation of the result There were two proofs of the above fact [Arv90], [Lie03] (also see [Arv03]), both of which involved substantial analysis. In a recent paper, Michael Skeide [Ske06] gave a more direct proof. In this note we present an elementary method for constructing an essential representation of any product system. Given the basic correspondence between E0semigroups and essential representations, the existence of an appropriate E0semigroup follows. Our terminology follows the monograph [Arv03]. Let E = {E(t) : t> 0} be a product system and choose a unit vector e ∈ E(1). e will be fixed throughout. We consider the Fréchet space of all Borel measurable sections t ∈ (0, ∞) ↦ → f(t) ∈ E(t) that are locally square integrable
Quantum Measurements of Scattered Particles
, 2015
"... We investigate the process of quantum measurements on scattered probes. Before scattering, the probes are independent, but they become entangled afterwards, due to the interaction with the scatterer. The collection of measurement results (the history) is a stochastic process of dependent random vari ..."
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We investigate the process of quantum measurements on scattered probes. Before scattering, the probes are independent, but they become entangled afterwards, due to the interaction with the scatterer. The collection of measurement results (the history) is a stochastic process of dependent random variables. We link the asymptotic properties of this process to spectral characteristics of the dynamics. We show that the process has decaying time correlations and that a zeroone law holds. We deduce that if the incoming probes are not sharply localized with respect to the spectrum of the measurement operator, then the process does not converge. Nevertheless, the scattering modifies the measurement outcome frequencies, which are shown to be the average of the measurement projection operator, evolved for one interaction period, in an asymptotic state. We illustrate the results on a truncated JaynesCummings model. 1 Introduction and main results We consider a scattering experiment in which a beam of probes is directed at a scatterer. The probes are sent to interact sequentially, one by one. Before the scattering process,