The $\Sigma$-QMAC problem is introduced, involving $S$ servers, $K$ classical ($\mathbb{F}_d$) data streams, and $T$ independent quantum systems. Data stream ${\sf W}_k, k\in[K]$ is replicated at a subset of servers $\mathcal{W}(k)\subset[S]$, and quantum system $\mathcal{Q}_t, t\in[T]$ is distributed among a subset of servers $\mathcal{E}(t)\subset[S]$ such that Server $s\in\mathcal{E}(t)$ receives subsystem $\mathcal{Q}_{t,s}$ of $\mathcal{Q}_t=(\mathcal{Q}_{t,s})_{s\in\mathcal{E}(t)}$. Servers manipulate their quantum subsystems according to their data and send the subsystems to a receiver. The total download cost is $\sum_{t\in[T]}\sum_{s\in\mathcal{E}(t)}\log_d|\mathcal{Q}_{t,s}|$ qudits, where $|\mathcal{Q}|$ is the dimension of $\mathcal{Q}$. The states and measurements of $(\mathcal{Q}_t)_{t\in[T]}$ are required to be separable across $t\in[T]$ throughout, but for each $t\in[T]$, the subsystems of $\mathcal{Q}_{t}$ can be prepared initially in an arbitrary (independent of data) entangled state, manipulated arbitrarily by the respective servers, and measured jointly by the receiver. From the measurements, the receiver must recover the sum of all data streams. Rate is defined as the number of dits ($\mathbb{F}_d$ symbols) of the desired sum computed per qudit of download. The capacity of $\Sigma$-QMAC, i.e., the supremum of achievable rates is characterized for arbitrary data replication and entanglement distribution maps $\mathcal{W}, \mathcal{E}$. Coding based on the $N$-sum box abstraction is optimal in every case. Notably, for every $S\neq 3$ there exists an instance of the $\Sigma$-QMAC where $S$-party entanglement is necessary to achieve the fully entangled capacity.

翻译：$\Sigma$-QMAC问题被引入，该问题涉及$S$个服务器、$K$个经典($\mathbb{F}_d$)数据流和$T$个独立量子系统。数据流${\sf W}_k, k\in[K]$被复制到服务器子集$\mathcal{W}(k)\subset[S]$中，量子系统$\mathcal{Q}_t, t\in[T]$则分布在服务器子集$\mathcal{E}(t)\subset[S]$中，使得服务器$s\in\mathcal{E}(t)$接收到子系统$\mathcal{Q}_{t,s}$（其中$\mathcal{Q}_t=(\mathcal{Q}_{t,s})_{s\in\mathcal{E}(t)}$）。服务器根据其数据操控各自的量子子系统，并将这些子系统发送给接收者。总下载代价为$\sum_{t\in[T]}\sum_{s\in\mathcal{E}(t)}\log_d|\mathcal{Q}_{t,s}|$量子比特，其中$|\mathcal{Q}|$表示$\mathcal{Q}$的维度。整个过程中要求在$t\in[T]$上$(\mathcal{Q}_t)_{t\in[T]}$的状态和测量是可分离的，但对于每个$t\in[T]$，$\mathcal{Q}_{t}$的子系统可以初始制备成任意（独立于数据的）纠缠态，由相应服务器任意操控，并由接收者联合测量。接收者必须从测量结果中恢复所有数据流的总和。速率定义为每量子比特下载中期望和的十进制符号（$\mathbb{F}_d$符号）数量。$\Sigma$-QMAC的容量（即可实现速率的 supremum）针对任意数据复制和纠缠分布映射$\mathcal{W}, \mathcal{E}$进行了刻画。基于$N$和盒抽象的编码在所有情况下都是最优的。值得注意的是，对于每个$S\neq 3$，存在一个$\Sigma$-QMAC的实例，其中需要$S$方纠缠才能实现完全纠缠容量。