Convergence Conditions for the Quantum Relative Entropy and Other Applications of the Deneralized Quantum Dini Lemma
Максим Евгеньевич Широков · Lobachevskii Journal of Mathematics · 2022
Abstract We describe a generalized version of the result called quantum Dini lemma that was used previously for analysis of local continuity of basic correlation and entanglement measures. The generalization consists in considering sequences of functions instead of a single function. It allows to expand the scope of possible applications of the method. We prove two general dominated convergence theorems and the theorem about preserving local continuity under convex mixtures. By using these theorems we obtain several convergence conditions for the quantum relative entropy and for the mutual information of a quantum channel considered as a function of a pair (channel, input state).