Theorem: Half-diamond information–disturbance bounds

For the complete interface $V$,

Equivalently,

Proof. In (eq:ksw-full), $q=2{\epsilon }_{\mathrm{rec}}(\mathcal{N})$ and $r=2{\delta }_{\mathrm{full}}(V)$. The left inequality becomes ${\epsilon }_{\mathrm{rec}}(\mathcal{N}{)}^2\le 2{\delta }_{\mathrm{full}}(V)$. The right becomes ${\delta }_{\mathrm{full}}(V)\le \sqrt{2{\epsilon }_{\mathrm{rec}}(\mathcal{N})}$. Rearrange and use the universal half-diamond upper bound one for a pair of channels.