Papers › Maximum and minimum entropy states yielding local continuity bounds
Maximum and minimum entropy states yielding local continuity bounds
Eric P. Hanson, Nilanjana Datta
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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.
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