Classical $(r,δ)$-locally recoverable codes are designed for avoiding loss of information in large scale distributed and cloud storage systems. We introduce the quantum counterpart of those codes by defining quantum $(r,δ)$-locally recoverable codes which are quantum error-correcting codes capable of correcting $δ-1$ qudit erasures from sets of at most $r+ δ-1$ qudits. We give a necessary and sufficient condition for a quantum stabilizer code $Q(C)$ to be $(r,δ)$-locally recoverable. Our condition depends only on the puncturing and shortening at suitable sets of both the symplectic self-orthogonal code $C$ used for constructing $Q(C)$ and its symplectic dual $C^{\perp_s}$. When $Q(C)$ comes from a Hermitian or Euclidean dual-containing code, and under an extra condition, we show that there is an equivalence between the classical and quantum concepts of $(r,δ)$-local recoverability. A Singleton-like bound is stated in this case and examples attaining the bound are given.

翻译：经典$(r,\delta)$局部可恢复码旨在避免大规模分布式及云存储系统中的信息丢失。我们通过定义量子$(r,\delta)$局部可恢复码引入这类码的量子对应物，这类量子纠错码能够从最多$r+\delta-1$个量子比特的集合中纠正$\delta-1$个量子比特擦除。我们给出了量子稳定子码$Q(C)$成为$(r,\delta)$局部可恢复码的充分必要条件。该条件仅取决于构造$Q(C)$所用的辛自正交码$C$及其辛对偶码$C^{\perp_s}$在特定集合上的收缩与缩短操作。当$Q(C)$源自埃尔米特或欧几里得对偶包含码，且在额外条件满足时，我们证明了$(r,\delta)$局部可恢复性的经典概念与量子概念之间存在等价关系。此时给出了类似Singleton界的界限，并提供了达到该界限的实例。