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, ₁ and 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 (₁ - ₂). This yields a novel entropic inequality that implies the well-known Audenaert-Fannes 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. Our proofs are largely based on majorization theory and convex optimization.