Close-to-optimal continuity bound for the von Neumann entropy and other quasi-classical applications of the Alicki-Fannes-Winter technique

We consider a quasi-classical version of the Alicki-Fannes-Winter technique widely used for quantitative continuity analysis of characteristics of quantum systems and channels. This version allows us to obtain continuity bounds under constraints of different types for quantum states belonging to subsets of a special form that can be called "quasi-classical". We describe several application of the proposed method. In particular, we obtain universal continuity bound for the von Neumann entropy under energy-type constraint which in the case of one-mode quantum oscillator is close to the specialized optimal continuity bound presented recently by Becker, Datta and Jabbour. It is essential that the proposed method applied to nonnegative functions gives one-side continuity bounds for quasi-classical states with rank/energy constraint imposed only on one state.