Quantum $(r,\delta)$-locally recoverable codes

Classical $(r,\delta)$-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 $(r,\delta)$-locally recoverable codes which are quantum error-correcting codes capable of correcting $\delta -1$ qudit erasures from sets of at most $r+ \delta -1$ qudits. We give a necessary and sufficient condition for a quantum stabilizer code $Q(C)$ to be $(r,\delta)$-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 $(r,\delta)$-local recoverability. A Singleton-like bound is stated in this case and examples attaining the bound are given.