Abstract We describe an advanced version of the AFW-technique proposed in [M. E. Shirokov, ‘‘Close-to-optimal continuity bound for the von Neumann entropy and other quasi-classical applications of the Alicki–Fannes–Winter technique’’, Lett. Math. Phys. 113, 121 (2023); M. E. Shirokov, ‘‘Local lower bounds on characteristics of quantum and classical systems’’, Lobachevskii J. Math. 44, 2169–2191 (2023)] which allows us to obtain lower semicontinuity bounds, continuity bounds and local lower bounds for characteristics of quantum systems and discrete random variables. We consider applications of the new version of the AFW-technique to several basic characteristics of quantum systems (the von Neumann entropy, the energy-type functionals, the quantum relative entropy, the conditional entropy and the entanglement of formation).
For the sandwiched Rényi entropy the conditional entropy can be defined two ways: Hα↓(A∣B)ρ,Hα↑(A∣B)ρ. In the limiting case, α=1, both definitions consolidate into conditional entropy H(A∣B)=S(AB)−S(B). The continuity inequality for conditional entropy H(A∣B), called the Alicki-Fannes-Winter (AWF) inequality, shows that if the states are close in trace-distance, then the conditional entropies are also close. Having the AWF inequality for conditional entropy, we show that the channel entropy defined through the relative entropy is continuous with respect to the diamond-distance between channels. Inspired by this, similar continuity inequalities for the Rényi conditional entropy Hα↑ were obtained in the work [A. Marwah and F. Dupuis, J. Math. Phys. 63, 052201 (2022)]. We provide continuity bounds for the sandwiched Rényi and Tsallis conditional entropies Hα↓(A∣B)ρ,Tα↓(A∣B)ρ for states with the same marginal on the conditioning system. Similar to the previous bounds, our bound depends only on the dimension of the conditioning system. We apply this result to prove continuity of the channel entropy for Rényi and Tsallis channel entropies defined through the sandwiched Rényi and Tsallis relative entropies.