Proposition: Additional interface error

If $\Vert \tilde{\Lambda }-\Lambda {\Vert }_{\diamond }\le \eta$, then

Combining state and interface perturbations yields

Proof. Use the reverse triangle inequality and the diamond-norm bound on $X_J$. Since $X_J$ is a difference of states, $\Vert X_J{\Vert }_1\le 2$. For the combined statement, first change the state at fixed $\tilde{\Lambda }$, then change the interface at fixed ideal state.