 Linearity of coherent quantum error correction

Suppose fixed unitary encoding and recovery correct each error $E_j$ by mapping every logical input $|\psi ⟩$ to $|\psi ⟩\otimes |{\eta }_j⟩$. 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 $E={\sum }_j{c}_j{E}_j$ is mapped to $|\psi ⟩\otimes {\sum }_j{c}_j|{\eta }_j⟩$. 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.