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

Abstract

The problem of entanglement-assisted summation over a quantum multiple access channel ($\Sigma$-QMAC) is intro-duced, involving$S$servers,$K$classical$(\mathbb{F}_{d})$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$\mathcal{Q}_{1}, \mathcal{Q}_{2}, \cdots, \mathcal{Q}_{S}$are prepared in advance and distributed to the corresponding servers. Each server$s, s\in[S]$locally manipulates its quantum system$\mathcal{Q}_{s}$according to its classical data and sends$\mathcal{Q}_{s}$to the receiver. The total communication cost is$\log_{d}\vert \mathcal{Q}_{1}\vert +\log_{d}\vert \mathcal{Q}_{2}\vert +\cdots+\log_{d}\vert \mathcal{Q}_{S}\vert$qudits, where$\vert \mathcal{Q}_{s}$denotes the dimension of$\mathcal{Q}_{s}$. Based on a measurement of the composite system$\mathcal{Q}_{1}\mathcal{Q}_{2}\cdots \mathcal{Q}_{S}$, the receiver must recover the desired sum. The rate thus achieved is defined as the number of dits$(\mathbf{F}_{d}$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$S, K, \mathbf{F}_{d}$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.