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On Extension of Quantum Channels and Operations to the Space of Relatively Bounded Operators

Lobachevskii Journal of Mathematics, ISSN: 1818-9962, Vol: 41, Issue: 4, Page: 714-727
2020
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Abstract: We analyse possibility to extend a quantum operation (sub-unital normal completely positive linear map on the algebra B(H) of bounded operators on a separable Hilbert space H to the space of all operators on H relatively bounded w.r.t. a given positive unbounded operator. We show that a quantum operation Φ can be uniquely extended to a bounded linear operator on the Banach space of all √G -bounded operators on H provided that the operation Φ is G-limited: the predual operation Φ maps the set of positive trace class operators ρ with finite value of TrρG into itself. Assuming that G has discrete spectrum of finite multiplicity we prove that for a wide class of quantum operations the existence of the above extension implies the G-limited property. Applications to the theory of Bosonic Gaussian channels are considered.

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