# On entanglement-assisted quantum codes achieving the entanglement-assisted Griesmer bound

On entanglement-assisted quantum codes achieving the entanglement-assisted Griesmer bound The theory of entanglement-assisted quantum error-correcting codes (EAQECCs) is a generalization of the standard stabilizer formalism. Any quaternary (or binary) linear code can be used to construct EAQECCs under the entanglement-assisted (EA) formalism. We derive an EA-Griesmer bound for linear EAQECCs, which is a quantum analog of the Griesmer bound for classical codes. This EA-Griesmer bound is tighter than known bounds for EAQECCs in the literature. For a given quaternary linear code $$\mathcal {C}$$ C , we show that the parameters of the EAQECC that EA-stabilized by the dual of $$\mathcal {C}$$ C can be determined by a zero radical quaternary code induced from $$\mathcal {C}$$ C , and a necessary condition under which a linear EAQECC may achieve the EA-Griesmer bound is also presented. We construct four families of optimal EAQECCs and then show the necessary condition for existence of EAQECCs is also sufficient for some low-dimensional linear EAQECCs. The four families of optimal EAQECCs are degenerate codes and go beyond earlier constructions. What is more, except four codes, our $$[[n,k,d_{ea};c]]$$ [ [ n , k , d e a ; c ] ] codes are not equivalent to any $$[[n+c,k,d]]$$ [ [ n + c , k , d ] ] standard QECCs and have better error-correcting ability than any $$[[n+c,k,d]]$$ [ [ n + c , k , d ] ] QECCs. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Quantum Information Processing Springer Journals

# On entanglement-assisted quantum codes achieving the entanglement-assisted Griesmer bound

, Volume 14 (12) – Oct 19, 2015
21 pages

/lp/springer_journal/on-entanglement-assisted-quantum-codes-achieving-the-entanglement-86Y8J3saUV
Publisher
Springer US
Subject
Physics; Quantum Information Technology, Spintronics; Quantum Computing; Data Structures, Cryptology and Information Theory; Quantum Physics; Mathematical Physics
ISSN
1570-0755
eISSN
1573-1332
D.O.I.
10.1007/s11128-015-1143-5
Publisher site
See Article on Publisher Site

### Abstract

The theory of entanglement-assisted quantum error-correcting codes (EAQECCs) is a generalization of the standard stabilizer formalism. Any quaternary (or binary) linear code can be used to construct EAQECCs under the entanglement-assisted (EA) formalism. We derive an EA-Griesmer bound for linear EAQECCs, which is a quantum analog of the Griesmer bound for classical codes. This EA-Griesmer bound is tighter than known bounds for EAQECCs in the literature. For a given quaternary linear code $$\mathcal {C}$$ C , we show that the parameters of the EAQECC that EA-stabilized by the dual of $$\mathcal {C}$$ C can be determined by a zero radical quaternary code induced from $$\mathcal {C}$$ C , and a necessary condition under which a linear EAQECC may achieve the EA-Griesmer bound is also presented. We construct four families of optimal EAQECCs and then show the necessary condition for existence of EAQECCs is also sufficient for some low-dimensional linear EAQECCs. The four families of optimal EAQECCs are degenerate codes and go beyond earlier constructions. What is more, except four codes, our $$[[n,k,d_{ea};c]]$$ [ [ n , k , d e a ; c ] ] codes are not equivalent to any $$[[n+c,k,d]]$$ [ [ n + c , k , d ] ] standard QECCs and have better error-correcting ability than any $$[[n+c,k,d]]$$ [ [ n + c , k , d ] ] QECCs.

### Journal

Quantum Information ProcessingSpringer Journals

Published: Oct 19, 2015

### References

• General entanglement-assisted quantum error-correcting codes
Hsieh, MH; Devetak, I; Brun, T
• High performance entanglement-assisted quantum LDPC codes need little entanglement
Hsieh, MH; Yen, WT; Hsu, LY
• Linear Plotkin bound for entanglement-assisted quantum codes
Guo, L; Li, R
• Quantum error correction via less noisy qubits
Fujiwara, Y
• Nonlocal quantum information in bipartite quantum error correction
Wilde, Mark M; Fattal, David

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