Continuity Bounds for Quantum Entropies Arising From a Fundamental Entropic Inequality

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.