Theorem: B3: intrinsic one-shot ledger for the authorized triple

Every CPTP exact recovery channel on $A=\{1,2,3\}$ equals $\mathcal{D}$ in (eq:D) on its entire input space. Its Choi rank is four.

In the pure-environment, one-shot dilation model, its minimum environment dimension is four. The output-plus-environment peak is three qubits, with no additional workspace beyond the batch input.

Proof. For another exact decoder ${\mathcal{D}}^{\prime }$, define $\mathcal{R}={\mathcal{D}}^{\prime }{\mathrm{Ad}}_{U^{†}}$. Equation (eq:normalform) and exact recovery give $\mathcal{R}(\rho \otimes I_4/4)=\rho$. Lemma lem:unique identifies $\mathcal{R}={\mathrm{Tr}}_F$, hence ${\mathcal{D}}^{\prime }=\mathcal{D}$.

The four Kraus maps

are Hilbert–Schmidt orthogonal: $\mathrm{tr}({K_f}^{†}{K}_g)=2{\delta }_{fg}$. Their Choi vectors are independent, so the Choi rank is four.

For completeness, an environment of dimension $r$ yields at most $r$ Kraus operators by expanding its dilation in a basis. Conversely, a minimum Kraus family defines the isometry $\psi \mapsto {\sum }_j{K}_j\psi \otimes |j⟩$. Thus the minimum environment dimension equals Choi rank Watrous (Chapter 2). Here the two environment qubits are the original syndrome factors, and the total output dimension is $2\cdot 4=8$.