Dualities and identities for entanglement-assisted quantum codes

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 (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. 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 PACS 03.67.-a · 03.67.Pp 2 errors acting on the n channel qubits,

Maximal entanglement entanglement-assisted quantum codes from quaternary BCH codes

“…An EAQECC, k/n is called its rate and (k − c)/n is called as its net rate. Let k = n − m. If c = k, then the EAQECC has parameters [[n, n − k, d; k]] and this EAQECC is called a maximal entanglement EAQECC in [8][9][10][11].…”

Entanglement-assisted quantum codes from arbitrary binary linear codes

“…Recently, Lai et al [19,20] proposed a new notion of maximal-entanglement entanglementassisted quantum error-correcting codes. From [19,20], we know that the one-shot father protocol is a random entanglement-assisted quantum code, and it achieves the entanglementassisted quantum capacity of a depolarizing channel by exploiting maximal entanglement.…”

“…From [19,20], we know that the one-shot father protocol is a random entanglement-assisted quantum code, and it achieves the entanglementassisted quantum capacity of a depolarizing channel by exploiting maximal entanglement. In addition, there is sufficient numerical evidence that maximal-entanglement turbo codes come within a few decibels of achieving the entanglement-assisted hashing bound.…”

The Road From Classical to Quantum Codes: A Hashing Bound Approaching Design Procedure

“…It is assumed furthermore that these pre-shared entangled qubits are transmitted over a noiseless quantum channel. The resultant EA hashing bound is given by [34], [48]:…”

“…For the family of maximally entangled codes associated with c = (n − k), the EA hashing bound of Eq. (5) is reduced to [34], [48]:…”