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

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,dea?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.