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 (
$\Sigma$
-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.