  Entanglement-assisted quantum error-correcting codes (EAQECCs) make use of
pre-existing entanglement between the sender and receiver to boost the rate of
transmission. It is possible to construct an EAQECC from any classical linear
code, unlike standard QECCs which can only be constructed from dual-containing
codes. Operator quantum error-correcting codes (OQECCs) allow certain errors to
be corrected (or prevented) passively, reducing the complexity of the
correction procedure. We combine these two extensions of standard quantum error
correction into a unified entanglement-assisted quantum error correction
formalism. This new scheme, which we call entanglement-assisted operator
quantum error correction (EAOQEC), is the most general and powerful quantum
error-correcting technique known, retaining the advantages of both
entanglement-assistance and passive correction. We present the formalism, show
the considerable freedom in constructing EAOQECCs from classical codes, and
demonstrate the construction with examples.
