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International Journal of Quantum Information

Maximal entanglement entanglement-assisted quantum codes of distance three

Luobin Guo, Qiang Fu, Ruihu Li, Liangdong Lü

Published Feb 1, 2015• 6 citations
Publisher

Abstract

Entanglement-assisted quantum error correcting code (EAQECC) is a generalization of standard stabilizer quantum code. Maximal entanglement EAQECCs can achieve the EA-hashing bound asymptotically. In this work, we give elementary recursive constructions of quaternary zero radical codes with dual distance three for all n ≥ 4. Consequently, good maximal entanglement EAQECCs of minimum distance three for such length n are obtained. Almost all of these EAQECCs are optimal or near optimal according to the EA-quantum Hamming bound.

DOI
10.1142/s0219749915500021

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Entanglement-assisted quantum error correcting code (EAQECC) is a generalization of standard stabilizer quantum code. Maximal entanglement EAQECCs can achieve the EA-hashing bound asymptotically. In this work, we give elementary recursive constructions of quaternary zero radical codes with dual distance three for all n ≥ 4. Consequently, good maximal entanglement EAQECCs of minimum distance three for such length n are obtained. Almost all of these EAQECCs are optimal or near optimal according to the EA-quantum Hamming bound.
Maximal entanglement entanglement-assisted quantum error correcting codes (EAQECCs) can achieve the EA-hashing bound asymptotically, and a higher rate and/or better noise suppression capabilities may be achieved by exploiting maximal entanglement. In this work, we discuss construction of some families of [n, k, d] zero radical codes with minimal distance d ≥ 6, 8, 10, 12, 14 from two classes of quaternary Bose-Chaudhuri-Hocquenghem (BCH) codes. Using these zero radical codes, we construct many maximal entanglement EAQECCs with very good parameters. We calculate the parameters of these EAQECCs by easily algebraical method, rather than by usually method of computation by computer. The maximal entanglement EAQECCs presented here have better parameters than those available in the literature.
The entanglement-assisted (EA) formalism is a generalization of the standard stabilizer formalism, and it can transform classical linear quaternary codes into entanglement-assisted quantum error correcting codes (EAQECCs) by using of shared entangled qubits between the sender and the receiver.In this work, we give elementary recursive constructions of special quaternary codes of length n and dual distance four that constructed from known caps in projective space PG(5,4) and PG(6,4) for all length 6n283.Consequently, good maximal entanglement EAQECCs of minimum distance four for such length n are constructed from the obtained quaternary codes.
Maximal-entanglement entanglement-assisted quantum error-correcting codes (EAQE-CCs) can achieve the EA-hashing bound asymptotically and a higher rate and/or better noise suppression capability may be achieved by exploiting maximal entanglement. In this paper, we discussed the construction of quaternary zero radical (ZR) codes of dimension five with length [Formula: see text]. Using the obtained quaternary ZR codes, we construct many maximal-entanglement EAQECCs with very good parameters. Almost all of these EAQECCs are better than those obtained in the literature, and some of these EAQECCs are optimal codes.
The entanglement-assisted (EA) formalism is a generalization of the standard stabilizer formalism, and it can transform arbitrary quaternary classical linear codes into entanglement-assisted quantum error correcting codes (EAQECCs) by using of shared entanglement between the sender and the receiver. Using a decomposition of the 126-cap in PG(5,4), we firstly constructed LCD n-cap with 6The entanglement-assisted (EA) formalism is a generalization of the standard stabilizer formalism, and it can transform arbitrary quaternary classical linear codes into entanglement-assisted quantum error correcting codes (EAQECCs) by using of shared entanglement between the sender and the receiver. Using a decomposition of the 126-cap in PG(5,4), we firstly constructed LCD n-cap with 6≤n≤120. From these LCD caps, we then derived the related maximal entanglement [[n, n-6,4;6]] EAQECCs. Finally, using LCD LCD subcaps of 126-cap obtained, we constructed maximal entanglement EAQECCs with parameters [[n, n-;k,4;k]] for 6≤k≤11.
Entanglement-assisted concatenated quantum codes (EACQCs) are constructed by concatenating two entanglement-assisted quantum error-correcting codes (EAQECCs). By selecting the inner and outer component codes carefully, it is able to construct state-of-the-art EACQCs with parameters better than previous quantum codes. In this work, we use almost maximum-distance-separable (MDS) codes and ℏ\hbarℏ-MDS codes as the outer codes to construct EACQCs. Because the range of code length of almost MDS and ℏ\hbarℏ-MDS codes is much more free than that of the commonly used MDS codes. We derive several families of new EACQCs with parameters better than the previously best known EAQECCs and standard quantum error-correcting codes (QECCs) of the same length and net transmissions. Moreover, we demonstrate that EACQCs are with maximal entanglement if both the inner and outer component codes are with maximal entanglement. As a result, we construct three new maximal-entanglement EACQCs which have optimal parameters. In addition, we present several new maximal-entanglement EACQCs whose minimum distance is only one less than the minimum distance of the optimal codes. In particular, we propose two new families of asymptotically good maximal-entanglement EACQCs with explicit constructions by using entanglement-assisted quantum algebraic geometry codes as the outer codes. At last, we prove that EACQCs can attain the quantum Gilbert-Varshamov bound for EAQECCs asymptotically.
The entanglement-assisted (EA) formalism is a generalization of the standard stabilizer formalism, and it can transform arbitrary quaternary classical linear codes into entanglement-assisted quantum error correcting codes (EAQECCs) by using of shared entanglement between the sender and the receiver. Using the special structure of linear EAQECCs, we derive an EA-Plotkin bound for linear EAQECCs, which strengthens the previous known EA-Plotkin bound. This linear EA-Plotkin bound is tighter then the EA-Singleton bound, and matches the EA-Hamming bound and the EA-linear programming bound in some cases. We also construct three families of EAQECCs with good parameters. Some of these EAQECCs saturate this linear EA-Plotkin bound and the others are near optimal according to this bound; almost all of these linear EAQECCs are degenerate codes.