  In the theory of operator quantum error correction (OQEC), the notion of
correctability is defined under the assumption that states are perfectly
initialized inside a particular subspace, a factor of which (a subsystem)
contains the protected information. If the initial state of the system does not
belong entirely to the subspace in question, the restriction of the state to
the otherwise correctable subsystem may not remain invariant after the
application of noise and error correction. It is known that in the case of
decoherence-free subspaces and subsystems (DFSs) the condition for perfect
unitary evolution inside the code imposes more restrictive conditions on the
noise process if one allows imperfect initialization. It was believed that
these conditions are necessary if DFSs are to be able to protect imperfectly
encoded states from subsequent errors. By a similar argument, general OQEC
codes would also require more restrictive error-correction conditions for the
case of imperfect initialization. In this study, we examine this requirement by
looking at the errors on the encoded state. In order to quantitatively analyze
the errors in an OQEC code, we introduce a measure of the fidelity between the
encoded information in two states for the case of subsystem encoding. A major
part of the paper concerns the definition of the measure and the derivation of
its properties. In contrast to what was previously believed, we obtain that
more restrictive conditions are not necessary neither for DFSs nor for general
OQEC codes. This is because the effective noise that can arise inside the code
as a result of imperfect initialization is such that it can only increase the
fidelity of an imperfectly encoded state with a perfectly encoded one.

Quantum Physics