Strong convergence of quantum channels: continuity of the Stinespring dilation and discontinuity of the unitary dilation

Abstract

We show that a sequence $\{{\Phi }_n\}$ of quantum channels strongly converges to a quantum channel ${\Phi }_0$ if and only if there exist a common environment for all the channels and a corresponding sequence $\{V_n\}$ of Stinespring isometries strongly converging to a Stinespring isometry $V_0$ of the channel ${\Phi }_0$. We also give a quantitative description of the above characterization of the strong convergence in terms of the appropriate metrics on the sets of quantum channels and Stinespring isometries. As a result, the uniform selective continuity of the complementary operation with respect to the strong convergence is established. We show discontinuity of the unitary dilation by constructing a strongly converging sequence of channels which can not be represented as a reduction of a strongly converging sequence of unitary channels. The Stinespring representation of strongly converging sequences of quantum channels allows to prove the lower semicontinuity of the entropic disturbance as a function of a pair (channel, input ensemble). Some corollaries of this property are considered.

23 pages, in v6 the main results are strengthened essentially, in v5 section 6 devoted to application of the main results in quantum information theory is added, while sections 6-7 of v4 are replaced to a separate paper (arXiv:1802.05632)

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• Lower semicontinuity of the relative entropy disturbance and its corollaries
• Compactness criterion for families of quantum operations in the strong convergence topology and its applications
• New monotonicity property of the quantum relative entropy