Maximum and minimum entropy states yielding local continuity bounds

Understand

Given an arbitrary quantum state ($\sigma$), we obtain an explicit construction of a state $\rho^*_\varepsilon(\sigma)$ (resp.

• $\rho_{*,\varepsilon}(\sigma)$) which has the maximum (resp.
• minimum) entropy among all states which lie in a specified neighbourhood ($\varepsilon$-ball) of $\sigma$.
• 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.