On the Capacity of Distributed Quantum Storage

A distributed quantum storage code maps a quantum message to N storage nodes, of arbitrary specified sizes, such that the stored message is robust to an arbitrary specified set of erasure patterns. The sizes of the storage nodes, and erasure patterns may not be homogeneous. The capacity of distributed quantum storage is the maximum feasible size of the quantum message (relative to the sizes of the storage nodes), when the scaling of the size of the message and all storage nodes by the same scaling factor is allowed. Representing the decoding sets as hyperedges in a storage graph, the capacity is characterized for various graphs, including MDS graph, wheel graph, Fano graph, and intersection graph. The achievability is related via quantum CSS codes to a classical secure storage problem. Remarkably, our coding schemes utilize non-trivial alignment structures to ensure recovery and security in the corresponding classical secure storage problem, which leads to similarly non-trivial quantum codes. The converse is based on quantum information inequalities, e.g., strong sub-additivity and weak monotonicity of quantum entropy, tailored to the topology of the storage graphs.