Abstract
Given an arbitrary quantum state (σ), we obtain an explicit construction of a state ρ^*_ε(σ) (resp. ρ_*,ε(σ)) which has the maximum (resp. minimum) entropy among all states which lie in a specified neighbourhood (ε-ball) of σ. Computing the entropy of these states leads to a local strengthening of the continuity bound of the von Neumann entropy, i.e., the Audenaert-Fannes inequality. Our bound is local in the sense that it depends on the spectrum of σ. The states ρ^*_ε(σ) and ρ_*,ε(σ) depend only on the geometry of the ε-ball and are in fact optimizers for a larger class of entropies. These include the R\'enyi entropy and the min- and max- entropies. This allows us to obtain local continuity bounds for these quantities as well. In obtaining this bound, we first derive a more general result which may be of independent interest, namely a necessary and sufficient condition under which a state maximizes a concave and G\^ateaux-differentiable function in an ε-ball around a given state σ. Examples of such a function include the von Neumann entropy, and the conditional entropy of bipartite states. Our proofs employ tools from the theory of convex optimization under non-differentiable constraints, in particular Fermat's Rule, and majorization theory.