Theorem: Dimension-explicit recovery–leakage bounds

For the complete finite interface,

The upper bound is an information-theoretic existence statement. It is not a resource certificate for an implemented decoder.

Proof. Fix any decoder $D$ and dilate it to $W:A\to C\otimes G$. Apply $(W\otimes I_B)V$ to one half of a Bell vector, producing a pure vector $\Psi$ on $Q,C,G,B$. Let $F_e$ be the decoded Bell overlap.

For $F_e>0$, project $\Psi$ onto ${\Phi }_d$ in $QC$ and normalize the environmental projection to $\chi$. The pure-state distance from ${\Phi }_d\otimes \chi$ is $\sqrt{1-F_e}$. Tracing $C,G$ gives some ${\tau }_B$ with

By Lemma lem:choipure,

If $F_e=0$, the same final inequality is trivial because $e(D\mathcal{N})=1$ and $\lambda \le 1$. Taking the infimum over $D$ proves the lower bound.

For the upper bound, choose a minimizing ${\tau }_B$ and put $\delta =\lambda (\mathcal{C})$. The Bell-input comparison gives

The second state has a purification ${\Phi }_d{}_{QC}\otimes {\chi }_{BG}$. The first has the purification obtained from $V$, with purifying system $A$. After enlarging $G$, purification alignment supplies an isometry $U:A\to C\otimes G$ whose aligned overlap with this product purification is at least $1-\delta$.

Define $D(Y)={\mathrm{Tr}}_G(UY{U}^{†})$. Its Bell overlap is at least $(1-\delta {)}^2$, because projection onto ${\Phi }_d$ includes the particular environmental vector $\chi$. Therefore

The constant-decoder alternative gives the other cap.