= Linearity of coherent quantum error correction
Suppose fixed unitary encoding and recovery correct each error $E_j$ by mapping every logical input $|\psi\rangle$ to $|\psi\rangle\otimes|\eta_j\rangle$. The syndrome vector can be taken independent of the logical input: applying linearity to basis inputs and their equal superposition forces their syndrome vectors to agree. Hence an error $E=\sum_jc_jE_j$ is mapped to $|\psi\rangle\otimes\sum_jc_j|\eta_j\rangle$. If $E$ is unitary, this final syndrome vector is normalized and the circuit corrects $E$. Orthogonality of the syndrome states is not needed for the linear-span argument; it is a separate condition concerning distinguishability of errors.
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