A Tight Uniform Continuity Bound for the Arimoto-Renyi Conditional Entropy and its Extension to Classical-Quantum States

We prove a tight uniform continuity bound for Arimoto's version of the conditional alpha-Renyi entropy for the range alpha is an element of [0, 1). This definition of the conditional alpha-Renyi entropy is the most natural one among the multiple forms which exist in the literature, since it satisfies two desirable properties of a conditional entropy, namely, the fact that conditioning reduces entropy, and that the associated reduction in uncertainty cannot exceed the information gained by conditioning. Furthermore, it has found interesting applications in various information theoretic tasks such as guessing with side information and sequential decoding. This conditional entropy reduces to the conditional Shannon entropy in the limit alpha -> 1, and this in turn allows us to recover the recently obtained tight uniform continuity bound for the latter from our result. Finally, we apply our result to obtain a tight uniform continuity bound for the conditional alpha-Renyi entropy of a classical-quantum state, for alpha in the same range as above. This again yields the corresponding known bound for the conditional entropy of the state in the limit alpha -> 1.

Automated summary

We prove a tight uniform continuity bound for Arimoto's version of the conditional alpha-Renyi entropy for the range alpha is an element of [0, 1). The conditional alpha-Renyi entropy is a natural and widely used concept in information theory, and it has applications in various tasks such as guessing and decoding. Our result also reveals the relationship between the conditional alpha-Renyi entropy and the conditional Shannon entropy.