We establish a tight upper bound for the difference in von Neumann entropies between two quantum states,$\rho _{1}$and$\rho _{2}$. This bound is expressed in terms of the von Neumann entropies of the mutually orthogonal states derived from the Jordan-Hahn decomposition of the difference operator$(\rho _{1} - \rho _{2})$. This yields a novel entropic inequality that implies the well-known Audenaert-Fannes (AF) inequality. In fact, it also leads to a refinement of the AF inequality. We employ this inequality to obtain a uniform continuity bound for the quantum conditional entropy of two states whose marginals on the conditioning system coincide. We additionally use it to derive a continuity bound for the quantum relative entropy in both variables. Interestingly, the fundamental entropic inequality is also valid in infinite dimensions.

This manuscript is a Correction to Becker et al., Trans. Inf. Theory 69, 4128 (2023). Specifically, we add some necessary assumptions inTheorems 2,4, and6therein, which were pointed out to us by Maksim Shirokov, and make some Corrections toFigure 2.

We prove a variety of improved uniform continuity bounds for entropies of both classical random variables on an infinite state space and of quantum states of infinite-dimensional systems. We obtain the first tight continuity estimate on the Shannon entropy of random variables with a countably infinite alphabet. The proof relies on a new mean-constrained Fano-type inequality. We then employ this classical result to derive a tight energy-constrained continuity bound for the von Neumann entropy. To deal with more general entropies in infinite dimensions,e.g.$\alpha $-Rényi and$\alpha $-Tsallis entropies, we develop a novel approximation scheme based on operator Hölder continuity estimates. Finally, we settle an open problem raised by Shirokov regarding the characterisation of states with finite entropy.

We prove a tight uniform continuity bound for Arimoto’s version of the conditional$\alpha $-Rényi entropy for the range$\alpha \in [0, 1$). This definition of the conditional$\alpha $-Rényi entropy is the most natural one among the multiple forms which exist in the literature, since it satisfies two desirable properties of a conditional entropy, namely, the fact that conditioning reduces entropy, and that the associated reduction in uncertainty cannot exceed the information gained by conditioning. Furthermore, it has found interesting applications in various information theoretic tasks such as guessing with side information and sequential decoding. This conditional entropy reduces to the conditional Shannon entropy in the limit$\alpha \to 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 $-Rényi entropy of a classical-quantum state, for$\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 \to 1$.