= Solution
A recovery operation must reverse every coherent superposition of the errors $\widetilde E_a$ without learning or disturbing the encoded state. Thus the corrupted subspaces associated with distinguishable syndromes must be orthogonal, while errors with the same syndrome must have identical action on the logical information. For any code states $|\psi\rangle,|\phi\rangle$, this means
$$
\langle\psi|\widetilde E_a^\dagger\widetilde E_b|\phi\rangle
=c_{ab}\langle\psi|\phi\rangle,
$$
with coefficients independent of the encoded states. In projector form these are exactly the <Knill--Laflamme condition>[Knill--Laflamme conditions]
$$
\boxed{\Pi_{\mathcal L}\widetilde E_a^\dagger\widetilde E_b\Pi_{\mathcal L}
=c_{ab}\Pi_{\mathcal L}.}
$$
Diagonalizing the positive matrix $c_{ab}$ chooses error combinations with mutually orthogonal syndrome spaces, which can be measured and reversed without revealing logical amplitudes.
Back to article page
