Proposition: Approximate leakage–recovery bounds

For Definition def:leakage,

Equivalently,

Proof. Use the dimension-independent continuity theorem for Stinespring dilations KSW. A complement at diamond distance $2{\delta }_{\mathrm{leak}}$ from a replacement channel admits aligned dilations at operator distance at most $\sqrt{2{\delta }_{\mathrm{leak}}}$. The comparison replacement dilation exposes an undisturbed copy of the input on its other output, yielding a decoder with half-diamond error at most $\sqrt{2{\delta }_{\mathrm{leak}}}$.

Conversely, dilate a decoder with recovered-channel error $2{\epsilon }_{\mathrm{rec}}$ and compare with a dilation of the identity. The complete complementary output is within half-diamond distance $\sqrt{2{\epsilon }_{\mathrm{rec}}}$ of a fixed state. Rearranging the two inequalities gives (eq:approximate).