= Necessity of the Knill-Laflamme condition
{title2=$PE_a^\dagger E_bP=\sum_\mu c_{\mu a}^*c_{\mu b}P$}
If a common recovery <quantum channel> with <Kraus operators> $R_\mu$ restores every pure code state after each error $E_a$, then each vector $R_\mu E_a|\psi\rangle$ is parallel to $|\psi\rangle$. <Linearity> on basis vectors and their pairwise superpositions forces $R_\mu E_aP=c_{\mu a}P$, with the scalar independent of the logical state. Insert $\sum_\mu R_\mu^\dagger R_\mu=I$ between two error operators to obtain the displayed <Knill--Laflamme condition>. This proves necessity without presuming an orthogonal syndrome description of recovery.
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