On the generators of quantum stochastic flows

Lindsay, J. Martin ; Parthasarathy, K. R. (1998) On the generators of quantum stochastic flows Journal of Functional Analysis, 158 (2). pp. 521-549. ISSN 0022-1236

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Official URL: http://linkinghub.elsevier.com/retrieve/pii/S00221...

Related URL: http://dx.doi.org/10.1006/jfan.1997.3194

Abstract

A time-indexed family of *-homomorphisms between operator algebras (jt:A→B)t∈I is called a stochastic process in quantum probability. When Ε C:B →C is a conditional expectation onto a subalgebra, the composed process (kt=C ° jt)t∈I is no longer *-homomorphic, but is completely positive and contractive. In some situations, the filtered process k may be described by a stochastic differential equation. The central aim of this paper is to study completely positive processes k which admit a differential description through a stochastic equation of the formed kt=kt°θαβ Λβα(t), in which Λ is the matrix of basic integrators of finite dimensional quantum stochastic calculus, and θ is a matrix of bounded linear maps on the algebra. The structure required of the matrix θ, for complete positivity of the process, is obtained. The stochastic generators of contractive, unital, and *-homomorphic processes are also studied. These results are applied to the equation dVt=lαβVtβα(t) in which l is a matrix of bounded Hilbert space operators.

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