Constructions of Good Entanglement-Assisted Quantum Error Correcting Codes

TL;DR

This paper explores the construction of entanglement-assisted quantum error correcting codes (EAQECCs), linking the required entanglement to classical code properties, and introduces methods for designing codes with optimal entanglement and error performance.

Contribution

It establishes a relationship between the entanglement needed and the hull of classical codes, providing new construction methods for EAQECCs with desirable entanglement and error correction capabilities.

Findings

Number of shared pairs relates to the classical code's hull.

Constructed families of EAQECCs with good error performance.

Proved existence of asymptotically good EAQECCs in odd characteristic.

Abstract

Entanglement-assisted quantum error correcting codes (EAQECCs) are a simple and fundamental class of codes. They allow for the construction of quantum codes from classical codes by relaxing the duality condition and using pre-shared entanglement between the sender and receiver. However, in general it is not easy to determine the number of shared pairs required to construct an EAQECC. In this paper, we show that this number is related to the hull of the classical code. Using this fact, we give methods to construct EAQECCs requiring desirable amount of entanglement. This leads to design families of EAQECCs with good error performance. Moreover, we construct maximal entanglement EAQECCs from LCD codes. Finally, we prove the existence of asymptotically good EAQECCs in the odd characteristic case.