Abstract
We introduce Wigner measures for infinite-dimensional open quantum systems; important examples of such systems are encountered in quantum control theory. In addition, we propose an axiomatic definition of coherent quantum feedback.
Highlights
We introduce Wigner measures for infinite-dimensional open quantum systems; important examples of such systems are encountered in quantum control theory
The Wigner measure is a generalization of the notion of the Wigner function, which was introduced by Wigner, and the representation of states of quantum systems in terms of Wigner measures is similar to the representation of states of classical Hamiltonian systems in terms of probability measures on the phase space
Describing the dynamics of open quantum systems in terms of Wigner measures allows one to apply methods similar to those used in describing the dynamics of open Hamiltonian systems
Summary
The Wigner measure is a generalization of the notion of the Wigner function, which was introduced by Wigner (see [5] and references therein), and the representation of states of quantum systems in terms of Wigner measures is similar to the representation of states of classical Hamiltonian systems in terms of probability measures on the phase space. If a Liouville measure does not exist, one can instead use a generalized measure invariant under the same symplectic transformations, which is naturally called a generalized Liouville measure; in this case, instead of the Wigner measure, one can consider its generalized density (cf [10]) with respect to the generalized Liouville measure. We consider the algebraic aspects of the theory and do not touch on assumptions of analytical character
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