Asymmetric Entanglement-Assisted Quantum Error-Correcting Codes and BCH Codes

Abstract

The concept of asymmetric entanglement-assisted quantum error-correcting code (asymmetric EAQECC) is introduced in this article. Codes of this type take advantage of the asymmetry in quantum errors since phase-shift errors are more probable than qudit-flip errors. Moreover, they use pre-shared entanglement between encoder and decoder to simplify the theory of quantum error correction and increase the communication capacity. Thus, asymmetric EAQECCs can be constructed from any pair of classical linear codes over an arbitrary field. Their parameters are described and a Gilbert-Varshamov bound is presented. Explicit parameters of asymmetric EAQECCs from BCH codes are computed and examples exceeding the introduced Gilbert-Varshamov bound are shown.2010 Mathematics Subject Classification. 81P70; 94B65; 94B05.

Improved Gilbert–Varshamov Bound for Entanglement-Assisted Asymmetric Quantum Error Correction by Symplectic Orthogonality

“…In this article, we propose a different version of the GV bound that uses the symplectic inner product and linear spaces, not necessarily direct products. Since our proposed GV bound covers a wider class of linear spaces, it gives a better existential bound than [5], as demonstrated by the examples in Table 1. Therefore, this article sheds new light on the optimal entanglement-assisted asymmetric quantum error correction.…”

“…Transactions on IEEE R. Matsumoto: Improved Gilbert-Varshamov Bound for Entanglement-Assisted Asymmetric Quantum Error Correction we compare our proposal with the conventional one [5]. Finally, concluding remarks are given in Section IV.…”

“…Galindo et al [4], [5] provided the following construction of entanglement-assisted asymmetric quantum error correction in their two subsequent papers.…”

“…Theorem 1 (see [4], [5]): Let C ⊆ F 2n q be an dimensional F q -linear space and H = (H X |H Z ) an × 2n matrix whose row space is C. Denote by c the minimum required number of maximally entangled quantum states in…”

“…On the other hand, until very recently, no researcher had studied entanglementassisted asymmetric quantum error correction, though its necessity and importance seem pretty obvious at hindsight. Galindo et al [5] recently introduced asymmetric error correction to EAQECCs, and also proved a Gilbert-Varshamov (GV) type existential theorem of codes with a given set of parameters. They used the direct products of two linear spaces and the Euclidean inner product.…”