Abstract
There exists a deep relationship between the nonexplosion conditions for Markov evolution in classical and quantum probability theories. Both of these conditions are equivalent to the preservation of the unit operator (total probability) by a minimal Markov semigroup. In this work, we study the Heisenberg evolution describing an interaction between the chain ofN two-level atoms andn-mode external Bose field, which was considered recently by J. Alli and J. Sewell. The unbounded generator of the Markov evolution of observables acts in the von Neumann algebra. For the generator of a Markov semigroup, we prove a nonexplosion condition, which is the operator analog of a similar condition suggested by R. Z. Khas’minski and later by T. Taniguchi for classical stochastic processes. For the operator algebra situation, this condition ensures the uniqueness and conservativity of the quantum dynamical semigroup describing the Markov evolution of a quantum system. In the regular case, the nonexplosion condition establishes a one-to-one relation between the formal generator and the infinitesimal operator of the Markov semigroup.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.