AI 中文总结
该研究针对量子局部可恢复码的约束限制，引入纠缠辅助量子局部可恢复码（EAQLRCs），建立其界与构造框架，得到首批明确的EAQLRCs最优码族。
AI 中文摘要
量子局部可恢复码（qLRCs）允许通过仅访问少量其他量子位来纠正单个量子位擦除。然而，标准CSS和埃尔米特构造对底层经典码施加对偶包含或自正交约束，从而限制了可用于构造qLRCs的结构良好的经典LRCs（cLRCs）。为了放松这些约束，我们引入纠缠辅助量子局部可恢复码（EAQLRCs），假设预共享最大纠缠对的一半是无噪声的。我们表征了扩展稳定子上的充分支撑条件，在此条件下纠缠辅助稳定子码具有局部性r，并从两个经典码推导出类似CSS的构造，而无需施加普通对偶包含条件。我们进一步为任意类似CSS的EAQLRCs建立了局部性上界和类Singleton界，并表征了达到后者界等式的纯码。这些结果提供了从cLRCs对构造最优纯EAQLRCs的通用框架。将该框架应用于MDS码的ℓ-交集对和块奇偶校验矩阵，我们获得了两个具有灵活参数和非平凡局部性的最优纯类似CSS的EAQLRCs族。据我们所知，这些是首批明确的EAQLRCs族。
英文摘要
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.
Comments22 pages, and another related work will be released soon
