The Capacity of Classical Summation over a Quantum MAC with Arbitrarily Replicated Inputs

The problem of entanglement-assisted summation over a quantum multiple access channel (S-QMAC) is introduced, involving S servers, K classical (Fd) data streams that are replicated arbitrarily across various subsets of servers, and a receiver who wishes to compute the sum of the K data streams. Independent of the data, entangled quantum systems Q(1), Q(2), center dot center dot center dot, Q(S) are prepared in advance and distributed to the corresponding servers. Each server s, s is an element of [S] locally manipulates its quantum system Q(s) according to its classical data and sends Q(s) to the receiver. The total communication cost is logd |Q(1)|+ logd |Q(2)|+ center dot center dot center dot + log(d) |Q(S)| qudits, where |Q(s)| denotes the dimension of Q(s). Based on a measurement of the composite system Q(1)Q(2) center dot center dot center dot Q(S), the receiver must recover the desired sum. The rate thus achieved is defined as the number of dits (F-d symbols) of the desired sum computed by the receiver per qudit (d-dimsional quantum system) of download. The capacity C is the supremum of the set of all achievable rates. As the main result of this work, the precise capacity of Sigma-QMAC is obtained, from which it follows that quantum entanglements allow a factor of 2 gain in capacity (superdense coding gain) relative to capacity with no entanglements, in all cases (any S, K, F-d and any data replication pattern) provided that the entanglement-assisted capacity does not exceed 1 dit/qudit (Holevo bound). Coding schemes based on a recent N-sum box abstraction are sufficient to achieve capacity.