Theorem: Recovery–leakage bounds with explicit constants

For the complete complementary channels of Definition def:interface,

Equivalently,

The corresponding symmetric bounds follow by exchanging ${ϵ}_{\mathrm{rec}}$ and ${\delta }_{\mathrm{leak}}$. In particular, ${ϵ}_{\mathrm{rec}}=0$ if and only if ${\delta }_{\mathrm{leak}}=0$.

Proof. Fix a state ${\tau }_F$. A dilation of its constant channel is

where ${\chi }_{\tau }$ purifies ${\tau }_F$. Applying Lemma lem:KSW to $\mathcal{C}$ and ${\mathcal{T}}_{\tau }$ supplies, after padding, an isometry $T:Z\to A\otimes G$ such that

Tracing $GF$, and using the right inequality of (eq:KSWraw), gives

Optimize over $\tau$.

Conversely, fix a decoder with error $ϵ$ and dilate it. The resulting channel is within raw diamond distance $2ϵ$ of the identity. Lemma lem:KSW aligns its dilation with $|\psi ⟩\mapsto |\psi ⟩\otimes |\chi ⟩$, with operator distance at most $\sqrt{2ϵ}$. Tracing the recovered logical output and the decoder environment therefore bounds the original complement by a constant channel with half-diamond error at most $\sqrt{2ϵ}$. Optimize over the decoder.

This reproduces the information–disturbance argument of KSW Theorem 3 with the two infima explicit. The factors of two arise solely from (eq:normalization); no square-root constant is guessed.