纠缠辅助量子(r,δ)本地可恢复码
Entanglement assisted quantum $(r,δ)$-locally recoverable codes
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中文总结 AI 辅助
本文提出纠缠辅助量子(r,δ)本地可恢复码框架，确立其性质的充要条件，建立相关概念联系与类Singleton界，并从多类经典码构造出最优的此类码。
中文摘要 AI 辅助
量子(r,δ)本地可恢复码是一类量子纠错码，能够在基数至多为r+δ-1的量子位（qudit）子集中纠正δ-1个量子位的擦除。本文引入更通用的纠缠辅助量子(r,δ)本地可恢复码框架，假设本地恢复操作由接收方持有的、不受擦除影响的量子位辅助。我们确立这类码满足该性质的充要条件；对源于Hermitian或Euclidean构造的码，建立纠缠辅助量子与经典(r,δ)本地可恢复性概念的联系，并推导类Singleton界；此外，从双变量J仿射簇码、BCH码、同比例变换BCH码等多类经典码构造最优纯纠缠辅助量子(r,δ)本地可恢复码。
英文摘要
Quantum $(r,δ)$-locally recoverable codes are quantum error-correcting codes capable of correcting $δ-1$ qudit erasures within one subset of qudits of cardinality at most $r+δ-1$. In this paper, we introduce the more general framework of entanglement-assisted quantum $(r,δ)$-locally recoverable codes, assuming that the local recovery operation is assisted by receiver-held qudits that remain unaffected by erasures. We establish necessary and sufficient conditions for these codes to satisfy this property. For codes derived from Hermitian or Euclidean constructions, we establish connections between entanglement-assisted quantum and classical notions of $(r,δ)$-local recoverability, and derive a Singleton-like bound. Furthermore, we construct optimal pure entan\-gle\-ment-assisted quantum $(r,δ)$-locally recoverable codes from several families of classical codes, including bivariate $J$-affine variety codes, BCH codes, and homothetic-BCH codes.
