Ching-Yi Lai, Todd A. Brun, Mark M. Wilde
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. For general EAQEC code, we derive linear programming bounds by exploiting these dualities and their corresponding MacWilliams identities. We also establish the Gilbert-Varshamov bound and the Plotkin bound for EAQEC codes, and use these to examine the existence of some EAQEC codes. Combining these bounds allows us to formulate a table of upper and lower bounds on the minimum distance of any maximal-entanglement EAQEC code with length up to 15 channel qubits. 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.
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http://arxiv.org/abs/1010.5506
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