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

Quantum Cross Subspace Alignment Codes via the $N$-sum Box Abstraction