 Necessity of the Knill-Laflamme condition

If a common recovery quantum channel with Kraus operators $R_{\mu }$ restores every pure code state after each error $E_a$, then each vector $R_{\mu }{E}_a|\psi ⟩$ is parallel to $|\psi ⟩$. Linearity on basis vectors and their pairwise superpositions forces $R_{\mu }{E}_{a}P=c_{\mu a}P$, with the scalar independent of the logical state. Insert ${\sum }_{\mu }{R}_{\mu }^{†}{R}_{\mu }=I$ between two error operators to obtain the displayed Knill--Laflamme condition. This proves necessity without presuming an orthogonal syndrome description of recovery.