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Dualities and identities for entanglement-assisted quantum codes

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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.

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Notes

  1. An \([[n, k, d;c]]\) EAQEC code is optimal in the sense that \(d\) is the highest achievable minimum distance for given parameters \(n\), \(k\), and \(c\).

  2. Poulin et al. described their decoder as a “maximum-likelihood” decoder [27], but a careful study of it reveals that their decoder should more properly be called a maximum a posteriori decoder.

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Author notes

  1. Mark M. Wilde

    Present address: Hearne Institute for Theoretical Physics, Center for Computation and Technology, Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA, 70803, USA

Authors and Affiliations

  1. Communication Sciences Institute, Electrical Engineering Department, University of Southern California, Los Angeles, CA, 90089, USA

    Ching-Yi Lai & Todd A. Brun

  2. School of Computer Science, McGill University, Montreal, QC, H3A 2A7, Canada

    Mark M. Wilde

Authors
  1. Ching-Yi Lai
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  2. Todd A. Brun
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  3. Mark M. Wilde
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Correspondence to Ching-Yi Lai.

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Lai, CY., Brun, T.A. & Wilde, M.M. Dualities and identities for entanglement-assisted quantum codes. Quantum Inf Process 13, 957–990 (2014). https://doi.org/10.1007/s11128-013-0704-8

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