Bounds on concatenated entanglement-assisted quantum error-correcting codes

TL;DR

This paper investigates bounds and properties of entanglement-assisted quantum error-correcting codes (EAQECCs), focusing on code concatenation, entanglement sharing, and error correction bounds, with new theoretical insights and constructions for non-maximal entanglement scenarios.

Contribution

It derives general formulas for entanglement sharing in concatenated EAQECCs, constructs families with order-independent entanglement, and extends bounds to non-maximal entanglement cases.

Findings

Derived the general expression for shared entanglement in concatenated codes.

Constructed families of EAQECCs with order-independent entanglement.

Extended bounds to non-maximal-entanglement EAQECCs.

Abstract

Code concatenation combines two or more component codes to design larger codes with greater noise resilience. Introducing entanglement assistance to concatenated codes provides a further advantage in terms of improved error rates and beating certain bounds on codes that would otherwise be unbeatable. First, we derive the general expression for the shared entanglement of a concatenated code and show that the number of ebits can depend on the order of concatenating the component entanglement-assisted quantum error-correcting codes (EAQECCs). We further construct families of pairs of EAQECCs such that the number of ebits of the resultant of concatenating the two codes in a given pair is order independent. Second, we derive conditions on code distance under which non-maximal-entanglement EAQECCs obtained from a classical quaternary Griesmer or Plotkin code saturate the entanglement-assisted (EA) Griesmer or linear EA Plotkin bound, respectively, extending the known result for maximal-entanglement EAQECCs. Furthermore, we present several families of such nonmaximal-entanglement EAQECCs. Third, we derive an EA version of the quantum Griesmer-Rains bound on the number of correctable errors for EAQECCs. Finally, we present families of pairs of EAQECCs such that the violation of the EA Hamming bound by the resultant of concatenating the two codes in a given pair is order dependent.