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

TL;DR

This paper investigates the conditions under which quantum channels converge strongly, establishing the continuity of Stinespring dilations, and demonstrates the discontinuity of unitary dilations through specific counterexamples.

Contribution

It provides a characterization of strong convergence of quantum channels via Stinespring isometries and explores the continuity properties of related quantum operations.

Findings

Strong convergence of channels corresponds to strong convergence of their Stinespring isometries.

The uniform selective continuity of the complementary operation is established.

Discontinuity of unitary dilation is demonstrated with explicit counterexamples.

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.