Entanglement-Assisted Quantum Locally Recoverable Codes: Characterizations, Bounds, and Constructions

2026-07-29

AI Summary

This work addresses the limitations of traditional quantum locally recoverable codes, which are constrained by dual-containing or self-orthogonality conditions that hinder full exploitation of classical code structures. The authors introduce an entanglement-assisted framework—the first of its kind for quantum locally recoverable codes—leveraging pre-shared noiseless entanglement to circumvent classical duality constraints. They establish a CSS-type construction based on two classical codes and, by integrating entanglement-assisted stabilizer code theory, ℓ-intersecting MDS codes, and block parity-check matrix techniques, derive upper bounds on locality and a Singleton-type bound. The paper characterizes pure codes achieving these bounds and explicitly constructs two families of optimal pure CSS codes with flexible parameters and nontrivial locality, thereby filling a critical gap in the explicit construction of optimal code families in this domain.

Abstract

Quantum locally recoverable codes (qLRCs) allow a single qudit erasure to be corrected by accessing only a small number of other qudits. Standard CSS and Hermitian constructions, however, impose dual-containing or self-orthogonal constraints on the underlying classical codes, thereby restricting the well-structured classical LRCs (cLRCs) that can be used to construct qLRCs. To relax these constraints, we introduce entanglement-assisted quantum locally recoverable codes (EAQLRCs) by assuming that halves of the pre-shared maximally entangled pairs are noiseless. We characterize sufficient support conditions on extended stabilizers under which entanglement-assisted stabilizer codes have locality $r$ and derive a CSS-like construction from two classical codes without imposing the ordinary dual-containing condition. We further establish an upper bound on locality and a Singleton-like bound for arbitrary CSS-like EAQLRCs, and characterize the pure codes attaining equality in the latter bound. These results yield a general framework for constructing optimal pure EAQLRCs from pairs of cLRCs. Applying this framework to $\ell$-intersection pairs of MDS codes and block parity-check matrices, we obtain two families of optimal pure CSS-like EAQLRCs with flexible parameters and nontrivial localities. To the best of our knowledge, these represent the first explicit families of EAQLRCs.

Problem

• quantum locally recoverable codes
• entanglement-assisted
• classical LRCs
• dual-containing constraint
• locality

Innovation

• entanglement-assisted
• quantum locally recoverable codes
• CSS-like construction
• Singleton bound
• MDS codes