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Given an arbitrary quantum state ($\sigma$), we obtain an explicitconstruction of a state $\rho^*_\varepsilon(\sigma)$ (resp.$\rho_{*,\varepsilon}(\sigma)$) which has the maximum (resp. minimum) entropyamong all states which lie in a specified neighbourhood ($\varepsilon$-ball) of$\sigma$. Computing the entropy of these states leads to a local strengtheningof the continuity bound of the von Neumann entropy, i.e., the Audenaert-Fannesinequality. Our bound is local in the sense that it depends on the spectrum of$\sigma$. The states $\rho^*_\varepsilon(\sigma)$ and$\rho_{*,\varepsilon}(\sigma)$ depend only on the geometry of the$\varepsilon$-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 allowsus to obtain local continuity bounds for these quantities as well. In obtainingthis bound, we first derive a more general result which may be of independentinterest, namely a necessary and sufficient condition under which a statemaximizes a concave and G\^ateaux-differentiable function in an$\varepsilon$-ball around a given state $\sigma$. Examples of such a functioninclude the von Neumann entropy, and the conditional entropy of bipartitestates. Our proofs employ tools from the theory of convex optimization undernon-differentiable constraints, in particular Fermat's Rule, and majorizationtheory.

Maximum and minimum entropy states yielding local continuity bounds