Continuity of characteristics of composite quantum systems
M. E. Shirokov
TL;DR
The paper develops two broad strategies for assessing continuity of state-characteristics in composite quantum systems: (i) the AFW framework and its refinements for uniform and local continuity, and (ii) an energy-constrained, purification-based approach that enables universal bounds via finite-energy approximations and purifications. These methods yield explicit, often asymptotically tight, bounds for central quantities such as the quantum conditional entropy, mutual information, classical correlation, quantum discord, and entanglement of formation, in both finite and infinite dimensions. By combining lower semicontinuity arguments and Dini-type convergence results, the authors derive robust local continuity criteria and demonstrate the applicability of the bounds to both standard finite-dimensional settings and energy-constrained infinite-dimensional regimes (e.g., oscillator models). The results have broad implications for quantum information processing, including channel capacities, information transmission under energy constraints, and stability analyses under perturbations, while also clarifying optimality and tightness in several regimes.
Abstract
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.