Entanglement assisted quantum $(r,\delta )$-locally recoverable codes

Abstract

Quantum $(r,\delta )$-locally recoverable codes are quantum error-correcting codes capable of correcting $\delta -1$ qudit erasures within one subset of qudits of cardinality at most $r+\delta -1$. In this paper, we introduce the more general framework of entanglement-assisted quantum $(r,\delta )$-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 $(r,\delta )$-local recoverability, and derive a Singleton-like bound. Furthermore, we construct optimal pure entan\-gle\-ment-assisted quantum $(r,\delta )$-locally recoverable codes from several families of classical codes, including bivariate $J$-affine variety codes, BCH codes, and homothetic-BCH codes.

cleveref.sty in arxiv.org TeXLive2025 works incorrectly and \Cref refers every Proposition, Remark etc. as Theorem. Choosing TeXLive2023 suppressed this issue, but the PDF could be rendered incorrectly when TeXLive 2023 is removed from arxiv.org