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arXiv:2608.17118

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

Carlos Galindo, Fernando Hernando, Helena Martín-Cruz, Ryutaroh Matsumoto

quant-phcs.ITSubmitted Aug 17, 2026• Announced Aug 19, 2026• 0 citations
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Abstract

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.

Comments
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

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Entanglement assisted quantum (r,δ)(r,δ)(r,δ)-locally recoverable codes
Quantum (r,δ)(r,δ)(r,δ)-locally recoverable codes are quantum error-correcting codes capable of correcting δ−1δ-1δ−1 qudit erasures within one subset of qudits of cardinality at most r+δ−1r+δ-1r+δ−1. In this paper, we introduce the more general framework of entanglement-assisted quantum (r,δ)(r,δ)(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)(r,δ)(r,δ)-local recoverability, and derive a Singleton-like bound. Furthermore, we construct optimal pure entan-gle-ment-assisted quantum (r,δ)(r,δ)(r,δ)-locally recoverable codes from several families of classical codes, including bivariate JJJ-affine variety codes, BCH codes, and homothetic-BCH codes.
Quantum locally recoverable codes (qLRCs) allow a single qudit erasure to be corrected by accessing only a small number of other qudits. Standard CSS and Hermitian constructions, however, impose dual-containing or self-orthogonal constraints on the underlying classical codes, thereby restricting the well-structured classical LRCs (cLRCs) that can be used to construct qLRCs. To relax these constraints, we introduce entanglement-assisted quantum locally recoverable codes (EAQLRCs) by assuming that halves of the pre-shared maximally entangled pairs are noiseless. We characterize sufficient support conditions on extended stabilizers under which entanglement-assisted stabilizer codes have locality rrr and derive a CSS-like construction from two classical codes without imposing the ordinary dual-containing condition. We further establish an upper bound on locality and a Singleton-like bound for arbitrary CSS-like EAQLRCs, and characterize the pure codes attaining equality in the latter bound. These results yield a general framework for constructing optimal pure EAQLRCs from pairs of cLRCs. Applying this framework to ℓ\ellℓ-intersection pairs of MDS codes and block parity-check matrices, we obtain two families of optimal pure CSS-like EAQLRCs with flexible parameters and nontrivial localities. To the best of our knowledge, these represent the first explicit families of EAQLRCs.
In this work, we define entanglement-assisted quantum locally recoverable codes with availability, in which any set of up to δ−1δ-1δ−1 erased qudits can be recovered from any one of ttt local recovery sets, each of size at most r+δ−1r+δ-1r+δ−1, with the recovery sets intersecting exactly in the erased coordinates, where rrr is a (small) positive integer. We show that shared entanglement permits t>1t>1t>1, meaning that multiple local recovery sets can be available for the same set of up to δ−1δ-1δ−1 erasures. We establish a Singleton-like bound for this family of codes and present random constructions based on classical linear codes with Vandermonde parity-check matrices. We also provide explicit constructions of entanglement-assisted quantum locally recoverable codes with availability from several classical code families and their folded versions, including Tamo-Barg codes, fiber-product codes, and algebraic-geometry codes such as one-point Hermitian and Suzuki codes.
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 ⊥ 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.
Bounds for General (r,δ)(r,δ)(r,δ)-Quantum Locally Recoverable Codes
We study general (r,δ)(r,δ)(r,δ)-quantum locally recoverable codes (qLRCs) without assuming purity or a stabilizer structure, and with arbitrarily overlapping recovery blocks. Using a selection lemma for multiple-erasure recovery, we establish a Singleton-like upper bound for general quantum subspaces. When δ=2δ=2δ=2, this bound reduces to the general single-erasure bound of Golowich and Guruswami. An integer optimization yields an explicit remainder correction that tightens the bound by up to δ−2δ-2δ−2. For pure codes, local decoupling and entropy arguments extend the parameter bound of Galindo et al., originally obtained for pure stabilizer codes constructed from Euclidean or Hermitian dual-containing classical codes, to arbitrary pure quantum subspaces. We also remove the partition assumption from several existing pure-code bounds for disjoint recovery blocks and obtain a further remainder correction by optimizing the budget. Finally, we analyze three families attaining these bounds: pure codes with overlapping recovery blocks and d=δd=δd=δ, which exhibit both a strict improvement of the pure-code bound and simultaneous equality in the pure-code and general bounds; impure codes with disjoint recovery blocks attaining the general bound; and an existing family of pure codes with overlapping recovery blocks and d>δd>δd>δ, which simultaneously attains all these bounds.
Locally recoverable codes (LRCs) permit reconstruction of a lost coordinate from a small subset of the remaining coordinates, a property that is a fundamental coding technique for distributed storage and, more recently, for fault-tolerant quantum memories and distributed storage systems. Existing constructions of quantum locally recoverable codes (qLRCs) and their entanglement-assisted generalizations (EA-qLRCs) impose a single locality parameter rrr governing recovery from both bit-flip (XXX-type) and phase-flip (ZZZ-type) errors, even though physical noise is rarely symmetric between the two. We introduce entanglement-assisted quantum locally recoverable codes (EA-qLRC) with \emph{asymmetric locality} {rX,rZ}\{r_X,r_Z\}{rX​,rZ​}, in which XXX-type and ZZZ-type errors are corrected from recovery sets of size at most rX+1r_X+1rX​+1 and rZ+1r_Z+1rZ​+1, respectively. We extend a stabilizer sufficient condition for this property and, in particular, the CSS framework, where asymmetric locality corresponds to separate locality conditions on the two classical codes C1\mathcal C_1C1​ and C2\mathcal C_2C2​ underlying the construction. From this characterization we derive Singleton-, Griesmer-, Plotkin-, sphere-packing-, and Cadambe\textendash Mazumdar-like bounds for CSS-type asymmetric EA-qLRCs (AEA-qLRCs) built from Proposition~\ref{CSS-EA-qLRC-Asym}, and we characterize exactly when a pure CSS-type AEA-qLRC built from a pair of classical LRCs attains the Singleton-like bound with equality.
For a general entanglement-assisted or unassisted quantum error-correcting code, a set of erasures, and an associated repair group, we propose a linear algebraic procedure to compute a reduced set of observables and a reduced repair group of codeword qudits for correcting the given erasures. The procedure has cubic complexity in the repair group size and the computed set of observables has the smallest possible size for correcting the given erasures. We specialize the general procedure to quantum codes constructed from Euclidean and Hermitian orthogonality, and provide closed-form upper bounds on the size of reduced repair groups for those special cases. Based on these closed-form bounds, we propose quantum counterparts of the information locality of classical local recovery.