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.

翻译：研究了在量子擦除多址接入信道（QEMAC）上计算分布式信源经典和的最优量子通信成本。K个由有限域符号组成的经典消息分布在S个服务器上，这些服务器预先共享量子纠缠。每个服务器s∈[S]根据其可用的经典消息操纵量子子系统Q_s，并将Q_s发送给接收端，接收端随后基于联合量子测量计算消息的和。来自服务器s∈[S]的下载成本是Q_s维度的对数。速率R定义为接收端计算出的和实例数除以所有服务器的总下载成本。主要关注对称场景，其中K = {S \choose \alpha}个消息，每个消息在唯一一组α个服务器间复制，且来自任意β个服务器的应答可能被擦除。若接收端最初无可用纠缠，我们证明容量（最大速率）恰为C= max\{min\{\frac{2(\alpha-\beta)}{S}, \frac{S-2\beta}{S}\}, \frac{\alpha-\beta}{S}\}。同时刻画了具有不同先验纠缠等级(Δ_0)的S个数据服务器与接收端之间的容量，通过引入一个无经典数据的辅助服务器（服务器0），使得来自服务器0的通信成本作为预先可用的接收端纠缠量的代理。逆问题的挑战在于弱单调性性质的最优应用，而可达性结合了经典网络编码和将量子比特视为经典dits的思想，以及基于N-sum盒抽象并依赖于绝对最大纠缠量子态的新构造。