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arXiv:1010.5506 · Quantum Information Processing

Dualities and identities for entanglement-assisted quantum codes

Ching-Yi Lai, Todd A. Brun, Mark M. Wilde

Published Dec 16, 2013• 37 citations
arXivHTMLPDF
DOI
10.1007/s11128-013-0704-8

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Abstract—The dual of an entanglement-assisted quantum error-correcting (EAQEC) code is the code resulting from ex-changing the original code’s information qubits for its ebits and vice versa. As an introduction to this notion, we show how entanglement-assisted repetition codes and entanglement-assisted 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 obtain linear programming bounds for EAQEC codes, by exploiting general forms of these dualities and corresponding MacWilliams identities. We establish the Gilbert-Varshamov bound and the Plotkin bound for EAQEC codes, and all of these bounds allow 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 an upper bound on the block error probability when transmitting maximal-entanglement EAQEC codes through the depolarizing channel. Index Terms—quantum dual code, entanglement-assisted quantum error correction, MacWilliams identity, linear programming bound, entanglement-assisted repetition codes, entanglement-assisted accumulator codes I.