General methods of quantitative and qualitative continuity analysis of
characteristics of composite quantum systems are described. Several
modifications of the Alicki-Fannes-Winter method are considered, which make it
applicable to a wide class of characteristics in both finite-dimensional and
infinite-dimensional cases. A new approximation method for obtaining local
continuity conditions for various characteristics of quantum systems is
proposed and described in detail. This method allows us to prove several
general results (Simon-type dominated convergence theorem, the theorem about
preserving continuity under convex mixtures, etc.).
Uniform continuity bounds and local continuity conditions for basic
characteristics of composite quantum systems are presented. Along with the
results obtained earlier by different authors, a number of new results proved
by the proposed methods are described.