Continuity inequalities for sandwiched Rényi and Tsallis conditional entropies with application to the channel entropy continuity

TL;DR

The paper develops uniform continuity bounds for sandwiched Rényi and Tsallis conditional entropies with fixed B marginals and uses them to prove continuity of corresponding channel entropies under diamond distance. It leverages the Alicki-Fannes-Winter framework by expressing channel entropy as an infimum over conditional entropies and extends known Rényi-continuity results with explicit, dimension-dependent bounds. The results provide quantitative stability guarantees for channel entropy measures defined via sandwiched divergences and highlight pseudo-additivity and normalization properties, with potential implications for channel capacity and related information-processing tasks. The work also indicates avenues for future study of non-sandwiched entropies and broader generalized divergences.

Abstract

For the sandwiched Rényi entropy the conditional entropy can be defined two ways: $\widetilde{H}^\downarrow_α(A|B)_ρ, \widetilde{H}^\uparrow_α(A|B)_ρ$. 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 $\widetilde{H}^\uparrow_α$ 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 $\widetilde{H}^\downarrow_α(A|B)_ρ, \widetilde{T}^\downarrow_α(A|B)_ρ$ 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.

Theorems & Definitions (21)

• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3
• proof
• Theorem 3.4
• proof
• Proposition 3.5
• proof