We develop the theory of entanglement-assisted quantum error correcting (EAQEC) codes, a generalization of the stabilizer formalism to the setting in which the sender and receiver have access to pre-shared entanglement. Conventional stabilizer codes are equivalent to dual-containing symplectic codes. In contrast, EAQEC codes do not require the dual-containing condition, which greatly simplifies their construction. We show how any quaternary classical code can be made into a EAQEC code. In particular, efficient modern codes, like LDPC codes, which attain the Shannon capacity, can be made into EAQEC codes attaining the hashing bound. In a quantum computation setting, EAQEC codes give rise to catalytic quantum codes which maintain a region of inherited noiseless qubits. We also give an alternative construction of EAQEC codes by making classical entanglement assisted codes coherent.

30 pages, 10 figures. Notation change: [[n,k;c]] instead of [[n,k-c;c]]