Information locality of a quantum locally recoverable code

Abstract

A classical linear code $C$ of length $n$ is said to have symbol locality $(r, δ)$ if for any index $j$ there exists a repair group $J_j \subseteq \{1, \ldots, n\}$ with $j\in J_j$ and $|J_j| \leq r+δ-1$ such that any $δ-1$ or fewer erasures in $J_j$ can be corrected by using codeword symbols only in $J_j$. Later it turned out that this way of defining $r$ overestimates the number of necessary codeword symbols for multiple-erasure correction, and information locality was proposed to define $r$ as the dimension of the punctured code of $C$ onto $J_j$. Recently locality $(r,δ)$ was proposed for quantum error-correcting codes by following the original definition of symbol locality $(r, δ)$. We propose a quantum counterpart of the information locality for quantum stabilizer codes constructed by Hermitian orthogonality, and a linear algebraic procedure computing a smaller repair group predicted by the proposed information locality and simultaneously reducing the number of measured observables in decoding to its minimum possible value. Then we demonstrate that the previously proposed definition of quantum locality $(r,δ)$ has the same drawback of overestimating the number of necessary codeword symbols for erasure correction by providing an explicit example of a quantum stabilizer code. Finally, we will give another example of a quantum stabilizer code constructed by Euclidean orthogonality and two different linear codes, with which a natural translation of the classical information locality into the quantum setting underestimates the number of necessary codeword symbols for erasure correction.