Theorem: Half-diamond recovery/leakage conversion

Under Convention conv:ksw,

In particular, if a specified state satisfies $d_{\diamond }(\mathcal{E},{\mathcal{T}}_{\sigma })\le \zeta$, then a deterministic decoder exists with error at most

Also,

Proof. By the definitions, $a=2{\epsilon }_{\mathrm{rec}}(\mathcal{N})$ and $b=2{\delta }_{\mathrm{leak}}(\mathcal{E})$. The first inequality of (eq:ksw-full) becomes ${{\epsilon }_{\mathrm{rec}}}^2\le 2{\delta }_{\mathrm{leak}}$. The second becomes ${\delta }_{\mathrm{leak}}\le \sqrt{2{\epsilon }_{\mathrm{rec}}}$. Lemma lem:constant-error gives the independent upper cap ${\epsilon }_{\mathrm{rec}}\le q_d$, and channel distances give ${\delta }_{\mathrm{leak}}\le 1$. A specified replacement allowance upper-bounds the minimized leakage. Squaring (eq:approx-reverse) proves (eq:leak-lower).