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The function described in the question is 1-Lipschitz. To argue this, we'll get an inequality in place before we start writing integrals. If $\vert \gamma\rangle$ and $\vert\delta\rangle$ are unit ve …

Let's start with the second question. There is nothing special about an extension $\sigma_{AR}^{\ast}$ that allows it to be optimal for the right-hand side of (1); any extension $\sigma_{AR}$ of $\sig …

Consider states $\rho_A$ and $\sigma_A$ that are close in $p$-norm (for $p>1$) but have relatively low fidelity. … The maximal fidelity between purifications $\Phi_{AB}$ and $\Psi_{AB}$ is also equal to $\delta$, so the minimal trace norm of the difference between purifications is bounded as follows: $$ \bigl\|\Phi …

The answer is no, as the following counter-example reveals. Let $\varepsilon\in(0,1)$ and define $$ \rho_0 = \begin{pmatrix} \frac{1+\varepsilon}{2} & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & \frac{1-\varepsilon …