Sufficient conditions for the conservativity of a minimal quantum dynamical semigroup
Let \({\mathcal H}\) be a separable Hilbert space with sesquilinear inner product \((\cdot,\cdot)\), and let \(\beta(H)\) be the von Neumann algebra that consists of all bounded operators \(H\) acting in \({\mathcal H}\). A formal generator \(L(\cdot)\) \[ L(B)= \phi(B)- \phi(I)\circ B+ i[H, B],\;\phi(B)= \sum_k \phi^*_k B\phi_k,\;A\circ B= \textstyle{{1\over 2}}(AB+ BA), \] of the quantum dynamical semigroup \(P_t(B)= e^{tL}(B)\) on the algebra \(\beta(H)\) is called regular if it determines the semigroup \(P_t(\cdot)\) uniquely. Assuming that the coefficients of \(L(\cdot)\) are a densely defined symmetric operator \(H\) acting in \({\mathcal H}\) (the Hamiltonian part), and \(\varphi\) is a normal completely positive map (the dissipative part) acting in \(\beta(H)\). The authors present sufficient conditions for a minimal quantum dynamical semigroup to be conservative. Moreover, the examples of equations in quantum optics are examined, most of which are regular.
scientific article; zbMATH DE number 1928212