Dualities and identities for entanglement-assisted quantum codes

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

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

23 Scopus citations

Abstract

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 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.

Original languageEnglish
Pages (from-to)957-990
Number of pages34
JournalQuantum Information Processing
Volume13
Issue number4
DOIs
StatePublished - 1 Jan 2014

Keywords

  • Entanglement-assisted accumulator codes
  • Entanglement-assisted quantum error correction
  • Entanglement-assisted repetition codes
  • Hashing bound
  • Linear programming bound
  • MacWilliams identity
  • Quantum dual code

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