Abstract
For an arbitrary uniformly continuous completely positive semigroup (ℱ t :t⩾0) on the space
of bounded operators on a Hilbert space
, we construct a family (U(t)∶t≥0) of unitary operators on a Hilbert space
and a conditional expectation
from
to
, such that, for arbitraryt≥0,
. The unitary operatorsU(t) satisfy a stochastic differential equation involving a noncommutative generalisation of infinite dimensional Brownian motion. They do not form a semigroup.
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References
Bratteli, O. and Robinson, D.:Operator Algebras and Statistical Mechanics, Volume II, Springer-Verlag, Berlin, 1981.
Hudson, R. L. and Parthasarathy, K. R.: ‘Quantumn Itô's Formula and Stochastic Evolutions’,Comm. Math. Phys. 93, (1984), 301–323.
Evans, D. and Lewis, J. T.:Dilations of Irreversible Evolutions in Algebraic Quantum Theory, Comm. Dublin Inst. for Adv. Studies, Series A24 (1974).
Lindblad, G.: ‘On the Generators of Quantum Dynamical Semigroups’Comm. Math. Phys. 48 (1946) 119–130.
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Part of this work was completed when the first author was visiting research associate at the Center for Relativity, Physics Department, The University of Texas at Austin, Austin, TX 78712, U.S.A., supported in part by NSF PHY 81-01381.
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Hudson, R.L., Parthasarathy, K.R. Stochastic dilations of uniformly continuous completely positive semigroups. Acta Appl Math 2, 353–378 (1984). https://doi.org/10.1007/BF02280859
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DOI: https://doi.org/10.1007/BF02280859