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In the theory of operator quantum error correction (OQEC), the notion ofcorrectability is defined under the assumption that states are perfectlyinitialized inside a particular subspace, a factor of which (a subsystem)contains the protected information. If the initial state of the system does notbelong entirely to the subspace in question, the restriction of the state tothe otherwise correctable subsystem may not remain invariant after theapplication of noise and error correction. It is known that in the case ofdecoherence-free subspaces and subsystems (DFSs) the condition for perfectunitary evolution inside the code imposes more restrictive conditions on thenoise process if one allows imperfect initialization. It was believed thatthese conditions are necessary if DFSs are to be able to protect imperfectlyencoded states from subsequent errors. By a similar argument, general OQECcodes would also require more restrictive error-correction conditions for thecase of imperfect initialization. In this study, we examine this requirement bylooking at the errors on the encoded state. In order to quantitatively analyzethe errors in an OQEC code, we introduce a measure of the fidelity between theencoded information in two states for the case of subsystem encoding. A majorpart of the paper concerns the definition of the measure and the derivation ofits properties. In contrast to what was previously believed, we obtain thatmore restrictive conditions are not necessary neither for DFSs nor for generalOQEC codes. This is because the effective noise that can arise inside the codeas a result of imperfect initialization is such that it can only increase thefidelity of an imperfectly encoded state with a perfectly encoded one.

Robustness of operator quantum error correction with respect to initialization errors