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 and entanglement distributions $\mathcal{W}, \mathcal{E}$. Coding based on the $N$-sum box abstraction is optimal in every case.

翻译：$\Sigma$-QMAC问题被提出，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$\mathcal{W}(k)\subset[S]$，量子系统$\mathcal{Q}_t, t\in[T]$被分布在服务器子集$\mathcal{E}(t)\subset[S]$中，使得服务器$s\in\mathcal{E}(t)$接收$\mathcal{Q}_t=(\mathcal{Q}_{t,s})_{s\in\mathcal{E}(t)}$的子系$\mathcal{Q}_{t,s}$。服务器根据其数据操控量子子系并将其发送给接收器。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$|\mathcal{Q}|$为$\mathcal{Q}$的维数。要求$(\mathcal{Q}_t)_{t\in[T]}$的状态和测量在$t\in[T]}$上始终可分离，但对于每个$t\in[T]$，$\mathcal{Q}_{t}$的子系可初始制备为任意（与数据无关的）纠缠态，由相应服务器任意操控，并由接收器联合测量。接收器必须从测量结果中恢复所有数据流之和。速率定义为每下载量子比特所计算的目标和的迪特（$\mathbb{F}_d$符号）数。对于任意数据与纠缠分布$\mathcal{W}, \mathcal{E}$，$\Sigma$-QMAC的容量（即可达速率的上确界）被完全刻画。基于$N$和盒抽象的编码方案在所有情况下均为最优。