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 $s\in[S]$ 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= {S \choose \alpha} $ 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 $R= \max\left\{ \min \left\{ \frac{2(\alpha-\beta)}{S}, 1-\frac{2\beta}{S} \right\}, \frac{\alpha-\beta}{S} \right\}$, which is shown to be optimal when $S\geq 2\alpha$.