Communication Efficiency of Summation over a Quantum Erasure MAC with Replicated Inputs

The quantum communication cost of computing a classical sum of distributed sources is studied over a quantum erasure multiple access channel (QEMAC).
$K$
- messages are distributed across
$S$
servers so that each server knows a subset of the messages. Each server
$\mathrm{s}\in[S]$
sends a quantum subsystem
$\mathcal{Q}_{s}$
to the receiver who computes the sum of the messages. The download cost from Server
$s\in[S]$
is the logarithm of the dimension of
$\mathcal{Q}_{s}$
. The rate
$R$
: is defined as the number of instances of the sum computed at the receiver, divided by the total download cost from all the servers. In the symmetric setting with
$K=\left(_{\alpha}^{S}\right)$
messages where each message is replicated among a unique subset of
$\alpha$
servers, and the answers from any
$\beta$
servers may be erased, the rate achieved is
$R=\max \left\{\min \left\{\frac{2(\alpha-\beta)}{S}, 1-\frac{2 \beta}{S}\right\}, \frac{\alpha-\beta}{S}\right\}$
.