Theorem: Exact recovery, dimension, and approximate correction

For a complete interface:

1. exact full-code recovery from $R$ is equivalent to $\mathcal{C}={\mathcal{T}}_{{\tau }_B}$;

2. if exact recovery holds and $s=\mathrm{rank}{\tau }_B$, then $\dim R\ge ds$;

3. $${\epsilon }_{\mathrm{rec}}(\mathcal{N})\le \min \{a_d,\sqrt{2{\delta }_{\mathrm{full}}(\mathcal{C})}\};$$

4. if $d_{\diamond }(\mathcal{N},{\mathcal{T}}_{{\omega }_R})\le \eta$, then

   $${\epsilon }_{\mathrm{rec}}(\mathcal{N})\ge \max \{0,a_d-\eta \}.$$

Proof. For exact recovery, dilate a decoder by $W:R\to L_{\mathrm{out}}G$. The composite $(W\otimes I_B)V$ dilates identity. The rank-one Choi operator of identity implies

for fixed $\chi$, proving complement constancy.

Conversely, diagonalize ${\tau }_B={{\sum }_{\alpha =1}}^s{\lambda }_{\alpha }|\alpha ⟩⟨\alpha |$ on its nonzero support. Write

Constancy on matrix units gives $⟨u_{j\beta }\mid u_{i\alpha }⟩={\delta }_{ij}{\delta }_{\alpha \beta }$. The resulting $ds$ orthonormal vectors define an isometry $T:L{\mathbb{C}}^s\to R$ and

The map

is CPTP and exactly recovers the code. This is the Knill–Laflamme structure KnillLaflamme.

The information–disturbance estimate gives

Changing normalization gives the square-root bound.

A decoded replacement has Bell overlap $d^{-2}$, hence half-diamond error at least $a_d$. Preparing $I_L/d$ attains this value: depolarization is a uniform average of $d^2$ Weyl conjugations, including identity. It also supplies the universal constant-decoder cap. Contraction and the triangle inequality prove (eq:replacementlower).