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 $r$ governing recovery from both bit-flip ( $X$ -type) and phase-flip ( $Z$ -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} $\{r_X,r_Z\}$ , in which $X$ -type and $Z$ -type errors are corrected from recovery sets of size at most $r_X+1$ and $r_Z+1$ , 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 $\mathcal C_1$ and $\mathcal C_2$ 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.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 $r$ governing recovery from both bit-flip ($X$-type) and phase-flip ($Z$-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} $\{r_X,r_Z\}$, in which $X$-type and $Z$-type errors are corrected from recovery sets of size at most $r_X+1$ and $r_Z+1$, 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 $\mathcal C_1$ and $\mathcal C_2$ 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.

quant-pharXiv:2610.23889Sep 29, 20260 citations

Asymmetric entanglement-assisted quantum codes: bound and constructions

Hualu Liu, Peng Hu, Xiusheng Liu

Designs Codes and CryptographyFeb 15, 20219 citations

r

governing recovery from both bit-flip (

X

-type) and phase-flip (

Z

-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}

\{r_X,r_Z\}

, in which

X

-type and

Z

-type errors are corrected from recovery sets of size at most

r_X+1

and

r_Z+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

\mathcal C_1

and

\mathcal C_2

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.