The Capacity of Classical Summation over a Quantum MAC with Arbitrarily Distributed Inputs and Entanglements
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The Σ-QMAC problem is introduced, involving S servers, K classical (𝔽_d) data streams, and T independent quantum systems. Data stream W_k, k∈[K] is replicated at a subset of servers 𝒲(k)⊂[S], and quantum system 𝒬_t, t∈[T] is distributed among a subset of servers ℰ(t)⊂[S] such that Server s∈ℰ(t) receives subsystem 𝒬_t,s of 𝒬_t=(𝒬_t,s)_s∈ℰ(t). Servers manipulate their quantum subsystems according to their data and send the subsystems to a receiver. The total download cost is ∑_t∈[T]∑_s∈ℰ(t)log_d|𝒬_t,s| qudits, where |𝒬| is the dimension of 𝒬. The states and measurements of (𝒬_t)_t∈[T] are required to be separable across t∈[T] throughout, but for each t∈[T], the subsystems of 𝒬_t can be prepared initially in an arbitrary (independent of data) entangled state, manipulated arbitrarily by the respective servers, and measured jointly by the receiver. From the measurements, the receiver must recover the sum of all data streams. Rate is defined as the number of dits (𝔽_d symbols) of the desired sum computed per qudit of download. The capacity of Σ-QMAC, i.e., the supremum of achievable rates is characterized for arbitrary data and entanglement distributions 𝒲, ℰ. Coding based on the N-sum box abstraction is optimal in every case.
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