Cross-subspace alignment (CSA) codes are used in various private information retrieval (PIR) schemes (e.g., with secure storage) and in secure distributed batch matrix multiplication (SDBMM). Using a recently developed $N$-sum box abstraction of a quantum multiple-access channel (QMAC), we translate CSA schemes over classical multiple-access channels into efficient quantum CSA schemes over a QMAC, achieving maximal superdense coding gain. Because of the $N$-sum box abstraction, the underlying problem of coding to exploit quantum entanglements for CSA schemes, becomes conceptually equivalent to that of designing a channel matrix for a MIMO MAC subject to given structural constraints imposed by the $N$-sum box abstraction, such that the resulting MIMO MAC is able to implement the functionality of a CSA scheme (encoding/decoding) over-the-air. Applications include Quantum PIR with secure and MDS-coded storage, as well as Quantum SDBMM.

翻译：交叉子空间对齐（CSA）码广泛应用于各类私有信息检索（PIR）方案（例如结合安全存储的场景）以及安全分布式批量矩阵乘法（SDBMM）中。借助近期提出的量子多址信道（QMAC）$N$-求和盒抽象，我们将经典多址信道上的CSA方案转化为QMAC上高效的量子CSA方案，从而实现了最大超密编码增益。由于$N$-求和盒抽象的存在，CSA方案中利用量子纠缠进行编码的核心问题在概念上等价于：在给定$N$-求和盒抽象所施加的结构约束条件下，为MIMO MAC设计信道矩阵，使得该MIMO MAC能够通过空中接口实现CSA方案的功能（编码/解码）。相关应用包括具有安全存储和MDS编码存储的量子PIR，以及量子SDBMM。