A quantum message is encoded into $N$ storage nodes (quantum systems $Q_1\dots Q_N$) with assistance from $N_B$ maximally entangled bi-partite quantum systems $A_1B_1, \dots, A_{N_B}B_{N_B}$, that are prepared in advance such that $B_1\dots B_{N_B}$ are stored separately as entanglement assistance (EA) nodes, while $A_1\dots A_{N_B}$ are made available to the encoder. Both the storage nodes and EA nodes are erasure-prone. The quantum message must be recoverable given any $K$ of the $N$ storage nodes along with any $K_B$ of the $N_B$ EA nodes. The capacity for this setting is the maximum size of the quantum message, given that the size of each EA node is $λ_B$. All node sizes are relative to the size of a storage node, which is normalized to unity. The exact capacity is characterized as a function of $N,K,N_B,K_B, λ_B$ in all cases, with one exception. The capacity remains open for an intermediate range of $λ_B$ values when a strict majority of the $N$ storage nodes, and a strict non-zero minority of the $N_B$ EA nodes, are erased. As a key stepping stone, an analogous classical storage (with shared-randomness assistance) problem is introduced. A set of constraints is identified for the classical problem, such that classical linear code constructions translate to quantum storage codes, and the converse bounds for the two settings utilize similar insights. In particular, the capacity characterizations for the classical and quantum settings are shown to be identical in all cases where the capacity is settled.

翻译：量子信息被编码至N个存储节点（量子系统Q₁…Q_N）中，并借助N_B个预先制备的最大纠缠双部量子系统A₁B₁, …, A_{N_B}B_{N_B}进行辅助，其中B₁…B_{N_B}作为纠缠辅助节点分别存储，而A₁…A_{N_B}可供编码器使用。存储节点与纠缠辅助节点均具有擦除倾向。量子信息必须能够在任意K个存储节点（共N个）与任意K_B个纠缠辅助节点（共N_B个）的条件下恢复。该场景下的容量定义为量子信息的最大可编码尺寸，其中每个纠缠辅助节点的尺寸为λ_B。所有节点尺寸均相对于归一化为单位的存储节点尺寸进行计算。除一种特殊情况外，容量在所有情况下均被精确表征为N、K、N_B、K_B、λ_B的函数。当N个存储节点中的严格多数以及N_B个纠缠辅助节点中的严格非零少数被擦除时，容量在λ_B的中间取值范围内仍为开放问题。作为关键理论基础，本文引入了具有共享随机性辅助的经典存储类比问题。通过识别经典问题的一组约束条件，经典线性码构造可转化为量子存储码，且两种场景的逆界证明运用了相似的思路。特别地，在容量已确定的所有情形中，经典与量子场景的容量表征被证明完全一致。