# Quantum Processes and Possibility of Their Control

## Abstract

The dissipation and decoherence (for example, the e®ects of noise in quantum computations), interaction with thermostat or in general with physical vacuum, measurement and many other complicated problems of open quantum systems are a consequence of interaction of quantum systems with the environment. These problems are described mathematically in terms of complex probabilistic process (CPP). Particularly, treating the environment as a Markovian process we derive an LangevinSchrÄodinger type stochastic di®erential equation (SDE) for describing the quantum system interacting with environment. For the 1D randomly quantum harmonic oscillator (QHO) L-Sh equation has a solution in the form of orthogonal CPP. On the basis of orthogonal CPP the stochastic density matrix (SDM) method is developed and in its framework relaxation processes in the uncountable dimension closed system of "QHO+environment" is investigated. With the help of SDM method the thermodynamical potentials, like nonequilibrium entropy and the energy of ground state are exactly constructed. The dispersion for di®erent operators is calculated. In particular, the expression for uncertain relations depending on parameter of interaction between QHO and environment is obtained. The Weyl transformation for stochastic operators is speci¯ed. Ground state Winger function is developed in detail.

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*Mathematical Problems of Computer Science*,

*26*, 101–113. Retrieved from http://mpcs.sci.am/index.php/mpcs/article/view/532

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