Capacity of Summation over a Symmetric Quantum Erasure MAC with Partially Replicated Inputs

The optimal quantum communication cost of computing a classical sum of distributed sources is studied over a quantum erasure multiple access channel (QEMAC). K classical messages comprised of finite-field symbols are distributed across $S$ servers, who also share quantum entanglement in advance. Each server $s\in[S]$ manipulates its quantum subsystem $\mathcal{Q}_s$ according to its own available classical messages and sends $\mathcal{Q}_s$ to the receiver who then computes the sum of the messages based on a joint quantum measurement. 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. The main focus is on 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. If no entanglement is initially available to the receiver, then we show that the capacity (maximal rate) is precisely $C= \max\left\{ \min \left\{ \frac{2(\alpha-\beta)}{S}, \frac{S-2\beta}{S} \right\}, \frac{\alpha-\beta}{S} \right\}$. The capacity with arbitrary levels of prior entanglement $(\Delta_0)$ between the $S$ data-servers and the receiver is also characterized, by including an auxiliary server (Server $0$) that has no classical data, so that the communication cost from Server $0$ is a proxy for the amount of receiver-side entanglement that is available in advance. The challenge on the converse side resides in the optimal application of the weak monotonicity property, while the achievability combines ideas from classical network coding and treating qudits as classical dits, as well as new constructions based on the $N$-sum box abstraction that rely on absolutely maximally entangled quantum states.