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Unitary Quantum Stochastic Evolutions
Author(s) -
Vincent-Smith G. F.
Publication year - 1991
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-63.2.401
Subject(s) - mathematics , unitary state , semigroup , pure mathematics , von neumann algebra , bounded function , space (punctuation) , von neumann architecture , quantum , class (philosophy) , conditional expectation , discrete mathematics , mathematical analysis , quantum mechanics , physics , linguistics , philosophy , political science , law , artificial intelligence , computer science , econometrics
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 L 1 , …, L 4 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 L 1 , …, L 4 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.