Continuity inequalities for sandwiched R\'enyi and Tsallis conditional entropies with application to the channel entropy continuity

Abstract

For the sandwiched R\'enyi entropy the conditional entropy can be defined two ways: $\widetilde{H}^\downarrow_\alpha(A|B)_\rho, \widetilde{H}^\uparrow_\alpha(A|B)_\rho$. In the limiting case, $\alpha=1$, 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 R\'enyi conditional entropy $\widetilde{H}^\uparrow_\alpha$ were obtained in the work [A. Marwah and F. Dupuis, J. Math. Phys. 63, 052201 (2022)]. We provide continuity bounds for the sandwiched R\'enyi and Tsallis conditional entropies $\widetilde{H}^\downarrow_\alpha(A|B)_\rho, \widetilde{T}^\downarrow_\alpha(A|B)_\rho$ 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 R\'enyi and Tsallis channel entropies defined through the sandwiched R\'enyi and Tsallis relative entropies.