Quantum Information Processing

, Volume 13, Issue 4, pp 957–990 | Cite as

Dualities and identities for entanglement-assisted quantum codes

Article

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.

Keywords

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

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Communication Sciences Institute, Electrical Engineering DepartmentUniversity of Southern CaliforniaLos AngelesUSA
  2. 2.School of Computer ScienceMcGill UniversityMontrealCanada
  3. 3.Hearne Institute for Theoretical Physics, Center for Computation and Technology, Department of Physics and AstronomyLouisiana State UniversityBaton RougeUSA

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