Proceedings of the London Mathematical Society · 1991 · 21 citations · 0 references
Spectral TheoryQuantum DynamicEngineeringContinuous SemigroupsFunctional AnalysisQuantum ComputingIntegrable ProbabilityVacuum Conditional ExpectationQuantum Mechanical PropertyQuantum EntanglementTopological SemigroupsQuantum SciencePhysicsStochastic Dynamical SystemFree ProbabilityNatural SciencesStochastic CalculusQuantum SystemPositive Semigroups
In the calculus of Hudson and Parthasarathy we show the existence of a solution to the quantum stochastic evolution equation U ( t ) = I + ∫ 0 t U ( s ) ( L 1 d Λ ( s ) + L 2 d A ( s ) + L 3 d A † ( s ) + L 4 d s ) for a class of unbounded operators L1, …, L4 in the initial space, and show that the necessary and sufficient condition that U be a unitary process is the same as in the case that L1, …, L4 are bounded. Applications of the vacuum conditional expectation to U give rise to strongly continuous semigroups and their generators in the initial space, and to ultraweakly continuous completely positive semigroups and their generators in the von Neumann algebra of the initial space.