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

2026-08-07

AI Summary

This work investigates the construction and performance limits of entanglement-assisted quantum locally recoverable codes (EA-qLRCs) without requiring the dual-containing condition. By leveraging pairs of classical locally recoverable codes and employing a CSS-like stabilizer framework, the authors introduce local recovery channels and incorporate hull dimension analysis to establish a unified upper-bound framework on code parameters. A general construction criterion is proposed that dispenses with the dual-containing requirement, yielding a tight Singleton-like bound together with necessary and sufficient conditions for its achievability. The study proves that cyclic codes can attain optimal EA-qLRCs, whereas Tamo–Barg codes achieve optimality only in degenerate cases. Furthermore, two Gilbert–Varshamov-type achievability bounds are derived for $q > 3$, fully characterizing the feasible region of rate, distance, and locality.

Abstract

This paper studies entanglement-assisted quantum locally recoverable codes (EA-qLRCs) built via a CSS-like stabilizer construction from pairs of classical locally recoverable codes (cLRCs), without requiring dual-containment. We define such codes through local recovery channels, give a sufficient stabilizer criterion for the construction, and derive Singleton-, Griesmer-, Plotkin-, and sphere-packing-like converse bounds on the parameters of the resulting pure CSS-like EA-qLRCs, along with a Cadambe--Mazumdar-like bound that, as in the classical case, lacks a closed form, plus a comparison of their relative tightness across finite-length and asymptotic regimes. We give necessary and sufficient conditions for a pure CSS-like EA-qLRC to attain the Singleton-like bound with equality; for the single-code case $\mathcal{C}_1=\mathcal{C}_2=\mathcal{C}$, this reduces to a simple condition on the hull dimension $s=\dim(\mathcal{C}\cap\mathcal{C}^\perp)$, which also fixes the entanglement count via $c=n-k-s$. We present CSS-like EA-qLRC constructions from classical LRC families---Tamo--Barg and cyclic codes---and characterize when these attain the Singleton-like bound, showing the cyclic families yield optimal codes while the Tamo--Barg construction, though valid, attains the bound only in the degenerate regime $k \le r$, where locality is vacuous. We complement these constructions with two Gilbert--Varshamov-like achievability bounds, via a classical parity-check augmentation and a sharper concatenated-code construction, and show both hold unconditionally for field size $q>3$ via a monomial-equivalence argument. Finally, we unify all bounds---converse and achievability alike---under a common maximally entangled regime, giving a single comparison of the achievable and forbidden rate--distance--locality region for CSS-like EA-qLRCs.

Problem

• entanglement-assisted quantum codes
• locally recoverable codes
• quantum coding bounds
• CSS construction
• code optimality

Innovation

• entanglement-assisted quantum codes
• locally recoverable codes
• CSS construction
• Singleton-like bound
• achievable rate region