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Strictly speaking, one should start with the Schrödinger-picture map T*, extend it by linearity to the trace class script T sign(H), and then define the Heisenberg-picture map T by noting that, for a fixed A ∈ B(H), Tr[T*(p)A] is a continuous linear functional on the trace class, and therefore has the form Tr(pA′) for some A′∈ B(H). We then set T(A) = A′. However, since we have assumed at the outset that T is normal, we do not have to worry too much about this nuance.
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This looks suspiciously like the Stinespring dilation of a CP map and, in fact, it is [for details see, e.g., Davies' proof (Ref. 2, Sec. 9.3) of the Naimark theorem]
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This looks suspiciously like the Stinespring dilation of a CP map and, in fact, it is [for details see, e.g., Davies' proof (Ref. 2, Sec. 9.3) of the Naimark theorem].
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