A stochastic Stinespring Theorem

Goswami, Debashish; Lindsay, J. Martin; Wills, Stephen J.

doi:10.1007/pl00004453

Cited by 10 publications

(10 citation statements)

References 15 publications

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“…The latter corresponds to the Evans-Hudson perturbation formula ( [17,13,19,4]) specialised to the case where the 'free' QS flow is implemented by a (Markov-regular) QS unitary cocycle V ; the perturbed QS flow is implemented by the unitary cocycle whose generator is (up to the choice of parameterisation) the series product of the stochastic generator of V and the perturbation coefficient.…”

Section: Proposition 44 Let V ∈ Qsmentioning

confidence: 99%

How to differentiate a quantum stochastic cocycle

Lindsay¹

2010

COSA

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Abstract. Two new approaches to the infinitesimal characterisation of quantum stochastic cocycles are reviewed. The first concerns mapping cocycles on an operator space and demonstrates the role of Hölder continuity; the second concerns contraction operator cocycles on a Hilbert space and shows how holomorphic assumptions yield cocycles enjoying an infinitesimal characterisation which goes beyond the scope of quantum stochastic differential equations.

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Section: Proposition 44 Let V ∈ Qsmentioning

confidence: 99%

How to differentiate a quantum stochastic cocycle

Lindsay¹

2010

COSA

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“…and similarly for the other processes. Under such assumptions (essentially bounded control requirements) we can show [BV06] that Z t is well defined and that U t has a unique unitary solution (see also [Mey93,Hol96,GLW01] for related results.) Before we discuss filtering in the context of Definition 3.1, let us clarify the significance of this definition.…”

Section: Definition 31 (Controlled Quantum Flow) Givenmentioning

confidence: 99%

On the Separation Principle in Quantum Control

Bouten¹,

Handel²

2008

Quantum Stochastics and Information

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It is well known that quantum continuous observations and nonlinear filtering can be developed within the framework of the quantum stochastic calculus of Hudson-Parthasarathy. The addition of real-time feedback control has been discussed by many authors, but the foundations of the theory still appear to be relatively undeveloped. Here we introduce the notion of a controlled quantum flow, where feedback is taken into account by allowing the coefficients of the quantum stochastic differential equation to be adapted processes in the observation algebra. We then prove a separation theorem for quantum control: the admissible control that minimizes a given cost function is a memoryless function of the filter, provided that the associated Bellman equation has a sufficiently regular solution. Along the way we obtain results on existence and uniqueness of the solutions of controlled quantum filtering equations and on the innovations problem in the quantum setting.

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“…The paper ends with a discussion of the perturbation of stochastic convolution cocycles by operator cocycles (cf. [9,14]). Perturbation of quantum Lévy processes by 'Weyl' cocycles corresponds at the generator level precisely to the action of the Euclidean group on Schürmann triples (cf.…”

Section: Introductionmentioning

confidence: 97%

Quantum stochastic convolution cocycles I

Lindsay

Skalski

2005

Annales de l'Institut Henri Poincare (B) Probability and Statis

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Stochastic convolution cocycles on a coalgebra are obtained by solving quantum stochastic differential equations. We describe a direct approach to solving such QSDE's by iterated quantum stochastic integration of matrix-sum kernels. The cocycles arising this way satisfy a Hölder condition, and it is shown that conversely every such Hölder-continuous cocycle is governed by a QSDE. Algebraic structure enjoyed by matrix-sum kernels yields a unital * -algebra of processes which allows easy deduction of homomorphic properties of cocycles on a 'quantum semigroup'. This yields a simple proof that every quantum Lévy process may be realised in Fock space. Finally perturbation of cocycles by Weyl cocycles is shown to be implemented by the action of the corresponding Euclidean group on Schürmann triples.  2005 Elsevier SAS. All rights reserved. RésuméDes cocycles de convolution stochastiques sur une coalgèbre sont obtenus par résolution des équations différentielles stochastiques (EDS) quantiques. Nous décrivons une méthode directe pour résoudre les EDS quantiques par intégration stochastique quantique itérée de noyaux matrice-somme. Les cocycles qui sont obtenus par cette méthode satisfont une condition de Hölder, et nous montrons réciproquement que chaque cocycle Hölder-continu est gouverné par une EDS quantique. La structure algé-brique des noyaux matrice-somme donne une *-algèbre de processus, qui nous permet une déduction facile des propriétés homomorphiques de cocycles sur un groupe quantique. Ce résultat permet d'obtenir un argument simple pour montrer que chaque processus de Lévy quantique peut être realisé dans l'espace de Fock. Finalement, nous montrons que la perturbation des cocycles par des cocycles de Weyl est mise en oeuvre par l'action du groupe euclidien correspondant sur les triplets de Schürmann.  2005 Elsevier SAS. All rights reserved.

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A stochastic Stinespring Theorem

Cited by 10 publications

References 15 publications

How to differentiate a quantum stochastic cocycle

How to differentiate a quantum stochastic cocycle

On the Separation Principle in Quantum Control

Quantum stochastic convolution cocycles I

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