纠缠辅助量子本地可恢复码：界、最优构造与可达性
Entanglement-Assisted Quantum Locally Recoverable Codes: Bounds, Optimal Constructions, and Achievability
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中文总结 AI 辅助
本文研究无需对偶包含的类CSS EA-qLRC，推导其各类参数界，给出等号达类Singleton界的条件，构造Tamo–Barg与循环码族的EA-qLRC，补充两个可达性界并统一所有界，明确循环码族可产生最优码。
中文摘要 AI 辅助
本文研究基于类CSS稳定子构造、由经典本地可恢复码（cLRC）对构建且无需对偶包含的纠缠辅助量子本地可恢复码（EA-qLRC）。我们通过本地恢复信道定义此类码，给出构造的充分稳定子判据，并推导所得纯类CSS EA-qLRC参数的类Singleton、类Griesmer、类Plotkin及类球填充逆界，以及与经典情况类似缺乏闭式的类Cadambe–Mazumdar界，同时对比其在有限长度与渐近区间内的相对紧度。我们给出纯类CSS EA-qLRC等号达到类Singleton界的充要条件；对于单码情况$\boldsymbol{\textit{C}}_1=\boldsymbol{\textit{C}}_2=\boldsymbol{\textit{C}}$，该条件简化为对核维数$s=\text{dim}(\boldsymbol{\textit{C}}\bigcap\boldsymbol{\textit{C}}^\bot)$的简单条件，且该条件还通过$c=n-k-s$确定纠缠数量。我们给出由经典LRC族——Tamo–Barg码与循环码——构造的类CSS EA-qLRC，并刻画这些码何时达到类Singleton界，结果显示循环码族可产生最优码，而Tamo–Barg构造虽有效，仅在$k \boldsymbol{\textit{\textless}} r$的退化区间（此时本地性无实际意义）达到该界。我们通过经典奇偶校验增广与更精细的级联码构造补充两个类Gilbert–Varshamov可达性界，并通过单项式等价论证证明这两个界在域大小$q\boldsymbol{\textit{\textgreater}}3$时无条件成立。最后，我们在统一的最大纠缠区间下整合所有逆界与可达性界，对类CSS EA-qLRC的可达与禁阻率-距离-本地性区域进行单一对比。
英文摘要
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.
