Theorem: Information–disturbance in half-diamond normalization

For Definition def:errors,

Proof. Fix a replacement channel whose distance from ${\mathcal{N}}^c$ is $\delta$. Its dilation is the embedding $|\psi ⟩\mapsto |\psi {⟩}_C|\chi {⟩}_{SK}$, where $\chi$ purifies the replacement state. The stated Stinespring comparison provides, after enlarging $S$, a decoder dilation bringing $V$ within operator norm $\sqrt{2\delta }$ of this embedding. Tracing its syndrome gives a decoded channel within half-diamond distance $\sqrt{2\delta }$ of ${\mathrm{id}}_C$. Minimize over replacement states.

Conversely, for a decoder of error $e$, compare a dilation of $\mathcal{DN}$ with a dilation of ${\mathrm{id}}_C$. Their operator-norm separation can be made at most $\sqrt{2e}$. Their complementary channels then differ by half-diamond distance at most $\sqrt{2e}$. Tracing the decoder syndrome leaves the original complete complement, while the identity dilation leaves a replacement channel. Minimize over decoders. The bound by one is automatic.