Unitary Quantum Stochastic Evolutions

Vincent-Smith, G. F.

doi:10.1112/plms/s3-63.2.401

Cited by 15 publications

(8 citation statements)

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“…The following Proposition extends the uniqueness theorems of Parthasarathy [21] Vincent-Smith [27] and Lindsay [15] to cmx quantum stochastic integrals. The integrand in (55) is zero almost everywhere for each f, g in the countable set L step Q .…”

Section: Processes Defined By Kernelssupporting

confidence: 56%

Theitô Formula for Quantum Semimartingales

Vincent-Smith¹

1997

Proceedings of the London Mathematical Society

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We construct a sequence of concordance invariants for classical links, which depend on the peripheral isomorphism type of the nilpotent quotients of the link fundamental group. The terminology stems from the fact that we replace the Magnus expansion in the definition of Milnor's μfalse¯‐invariants by the similar Campbell–Hausdorff expansion. The main point is that we introduce a new universal indeterminacy, which depends only on the number of components of the link. The Campbell–Hausdorff invariants are new, effectively computable and can efficiently distinguish (unordered and unoriented) isotopy types of links, as we indicate on several families of closed braid examples. They also satisfy certain natural dependence relations, which generalize well‐known symmetries of the μfalse¯‐invariants. 1991 Mathematics Subject Classification: 81S25, 46L10, 46L50, 47A60.

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Section: Processes Defined By Kernelssupporting

confidence: 56%

Theitô Formula for Quantum Semimartingales

Vincent-Smith¹

1997

Proceedings of the London Mathematical Society

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“…Furthermore it is an injection, because if F = 0, then X = 0, as F − − = X = F + + . Conversely, if X = 0, then t 0 F µ ν d A ν µ = 0 for all t, but it implies F µ ν = 0 due to the independence of stochastic integrators [15]. Now let us consider an adapted selfadjoint QS process Y , satisfying a QS equation…”

Section: Qs Calculus Of Output Nondemolition Processesmentioning

confidence: 99%

Quantum stochastic calculus and quantum nonlinear filtering

Belavkin¹

1992

Journal of Multivariate Analysis

307

368

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A ⋆-algebraic indefinite structure of quantum stochastic (QS) calculus is introduced and a continuity property of generalized nonadapted QS integrals is proved under the natural integrability conditions in an infinitely dimensional nuclear space. The class of nondemolition output QS processes in quantum open systems is characterized in terms of the QS calculus, and the problem of QS nonlinear filtering with respect to nondemolition continuous measurments is investigated. The stochastic calculus of a posteriori conditional expectations in quantum observed systems is developed and a general quantum filtering stochastic equation for a QS process is derived. An application to the description of the spontaneous collapse of the quantum spin under continuous observation is given.

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“…An extension allowing the inclusion of classical Markov chains with infinite state space was done in [10]. Another extension to quantum processes realised by conjugations with respect to a unitary operator-valued adapted process was given in [2] (see also [3,14]) as a corollary of an existence uniqueness and unitarity theorem on quantum stochastic differential equations with unbounded coefficients. The above results, however, cannot be applied to obtain a realization as quantum flow of the most important class of classical diffusions, Ito diffusions on U N .…”

Section: Introductionmentioning

confidence: 99%