Sufficient conditions for the conservativity of a minimal quantum dynamical semigroup
type: Item
instance of: scholarly article
title: Sufficient conditions for the conservativity of a minimal quantum dynamical semigroup
zbMATH Open document ID: 1024.37006
DOI: 10.1023/A:1015896123403
author: S. Yu. Shustikov
author: Alexander M. Chebotarev
published in: Mathematical Notes
publication date: 2003-06-15T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 37B05
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 47N50
Mathematics Subject Classification ID: 81S25
Mathematics Subject Classification ID: 81V80
zbMATH DE Number: 1928212
zbMATH Keywords: minimality
zbMATH Keywords: von Neumann algebra
zbMATH Keywords: quantum dynamical semigroup
zbMATH Keywords: symmetric operator
zbMATH Keywords: quantum optics
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
full work available at URL: a:1015896123403
OpenAlex ID: W146392848
description: scientific article; zbMATH DE number 1928212