Capacity of Summation over a Symmetric 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). $K$ classical messages are distributed across $S$ servers, who also share quantum entanglement in advance. Each server $s\in[S]$ manipulates and sends its 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, we show that the capacity (maximal rate) is $C= \max\left\{ \min \left\{ \frac{2(\alpha-\beta)}{S}, \frac{S-2\beta}{S} \right\}, \frac{\alpha-\beta}{S} \right\}$.