Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 342 / 2 / a / Solution

A recovery operation must reverse every coherent superposition of the errors ${\tilde{E}}_a$ 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 $|\psi ⟩,|\varphi ⟩$, this meanswith coefficients independent of the encoded states. In projector form these are exactly the Knill--Laflamme conditionsDiagonalizing the positive matrix $c_{ab}$ chooses error combinations with mutually orthogonal syndrome spaces, which can be measured and reversed without revealing logical amplitudes.