Quantum error-correcting subsystems are unitarily recoverable subsystems

We show that every correctable subsystem for an arbitrary noise operation can be recovered by a unitary operation, where the notion of recovery is more relaxed than the notion of correction insofar as it does not protect the subsystem from subsequent iterations of the noise. We also demonstrate that in the case of unital noise operations one can identify a subset of all correctable subsystems—those that can be corrected by a single unitary operation—as the noiseless subsystems for the composition of the noise operation with its dual. Using the recently developed structure theory for noiseless subsystems, the identification of such unitarily correctable subsystems is reduced to an algebraic exercise.