  The dual of an entanglement-assisted quantum error-correcting (EAQEC) code is
the code resulting from exchanging the original code's information qubits with
its ebits. To introduce this notion, we show how entanglement-assisted (EA)
repetition codes and accumulator codes are dual to each other, much like their
classical counterparts, and we give an explicit, general quantum shift-register
circuit that encodes both classes of codes.We later show that our constructions
are optimal, and this result completes our understanding of these dual classes
of codes. We also establish the Gilbert-Varshamov bound and the Plotkin bound
for EAQEC codes, and we use these to examine the existence of some EAQEC codes.
Finally, we provide upper bounds on the block error probability when
transmitting maximal-entanglement EAQEC codes over the depolarizing channel,
and we derive variations of the hashing bound for EAQEC codes, which is a lower
bound on the maximum rate at which reliable communication over Pauli channels
is possible with the use of pre-shared entanglement.
