Compactness criterion for families of quantum operations in the strong convergence topology and its applications

TL;DR

This work refines the compactness criterion for families of quantum operations in the strong convergence topology by leveraging the generalized Choi–Jamiołkowski isomorphism and provides a detailed proof including the previously missing closedness condition. It develops equivalent conditions for the existence of limit points of sequences of quantum operations and derives broad applications, including compactness of input-output constraint sets, energy-bounded channels, and robust limit-point criteria. The authors extend fundamental results to infinite dimensions, notably Petz's theorem for non-faithful states, reversibility preservation under strong convergence, and the existence of Fawzi–Renner recovery channels with extended QCMI, while also showing closedness of degradable/anti-degradable channels and continuity of quantum relative entropy under strong convergence. Together, these results supply a powerful framework for analyzing infinite-dimensional quantum channels and their limiting behavior, with implications for quantum information theory in continuous-variable and other infinite-dimensional settings.

Abstract

A revised version of the compactness criterion for families of quantum operations in the strong convergence topology (obtained previously) is presented, along with a more detailed proof and the examples showing the necessity of this revision. Several criteria for the existence of a limit point of a sequence of quantum operations w.r.t. the strong convergence are obtained and discussed. Applications in different areas of quantum information theory are described.