Entanglement-Assisted Quantum Error-Correcting Codes over Local Frobenius Rings

TL;DR

This paper introduces a new framework for constructing entanglement-assisted quantum error-correcting codes from classical additive codes over local Frobenius rings, extending existing results and providing explicit formulas for code parameters.

Contribution

It develops a standard form construction for additive codes over Frobenius rings and derives exact formulas for entanglement requirements, advancing quantum coding theory over rings.

Findings

Provides a standard form for additive codes over Frobenius rings.

Derives exact entanglement requirements for Galois ring-based codes.

Shows how adding coordinates influences code parameters.

Abstract

In this paper, we provide a framework for constructing entanglement-assisted quantum error-correcting codes (EAQECCs) from classical additive codes over a finite commutative local Frobenius ring $\mathcal{R}$. At the heart of the framework, and this is one of the main technical contributions of our paper, is a procedure to construct, for an additive code $\mathcal{C}$ over $\mathcal{R}$, a generating set for $\mathcal{C}$ that is in standard form, meaning that it consists purely of isotropic generators and hyperbolic pairs. Moreover, when $\mathcal{R}$ is a Galois ring, we give an exact expression for the minimum number of pairs of maximally entangled qudits required to construct an EAQECC from an additive code over $\mathcal{R}$, which significantly extends known results for EAQECCs over finite fields. We also demonstrate how adding extra coordinates to an additive code can give us a certain degree of flexibility in determining the parameters of the EAQECCs that result from our construction.