The $N$-sum box protocol specifies a class of $\mathbb{F}_d$ linear functions $f(W_1,\cdots,W_K)=V_1W_1+V_2W_2+\cdots+V_KW_K\in\mathbb{F}_d^{m\times 1}$ that can be computed at information theoretically optimal communication cost (minimum number of qudits $\Delta_1,\cdots,\Delta_K$ sent by the transmitters Alice$_1$, Alice$_2$,$\cdots$, Alice$_K$, respectively, to the receiver, Bob, per computation instance) over a noise-free quantum multiple access channel (QMAC), when the input data streams $W_k\in\mathbb{F}_d^{m_k\times 1}, k\in[K]$, originate at the distributed transmitters, who share quantum entanglement in advance but are not otherwise allowed to communicate with each other. In prior work this set of optimally computable functions is identified in terms of a strong self-orthogonality (SSO) condition on the transfer function of the $N$-sum box. In this work we consider an `inverted' scenario, where instead of a feasible $N$-sum box transfer function, we are given an arbitrary $\mathbb{F}_d$ linear function, i.e., arbitrary matrices $V_k\in\mathbb{F}_d^{m\times m_k}$ are specified, and the goal is to characterize the set of all feasible communication cost tuples $(\Delta_1,\cdots,\Delta_K)$, not just based on $N$-sum box protocols, but across all possible quantum coding schemes. As our main result, we fully solve this problem for $K=3$ transmitters ($K\geq 4$ settings remain open). Coding schemes based on the $N$-sum box protocol (along with elementary ideas such as treating qudits as classical dits, time-sharing and batch-processing) are shown to be information theoretically optimal in all cases. As an example, in the symmetric case where rk$(V_1)$=rk$(V_2)$=rk$(V_3) \triangleq r_1$, rk$([V_1, V_2])$=rk$([V_2, V_3])$=rk$([V_3, V_1])\triangleq r_2$, and rk$([V_1, V_2, V_3])\triangleq r_3$ (rk = rank), the minimum total-download cost is $\max \{1.5r_1 + 0.75(r_3 - r_2), r_3\}$.

翻译：$N$-和箱协议规定了一类$\mathbb{F}_d$线性函数$f(W_1,\cdots,W_K)=V_1W_1+V_2W_2+\cdots+V_KW_K\in\mathbb{F}_d^{m\times 1}$，这些函数可以在无噪声量子多址信道（QMAC）上以信息论最优通信成本（每个计算实例中发送方Alice$_1$, Alice$_2$,$\cdots$, Alice$_K$分别向接收方Bob发送的最小量子比特数$\Delta_1,\cdots,\Delta_K$）进行计算，此时输入数据流$W_k\in\mathbb{F}_d^{m_k\times 1}, k\in[K]$来源于分布式发送方，这些发送方预先共享量子纠缠但除此之外不允许彼此通信。在先前工作中，这类最优可计算函数通过$N$-和箱传递函数的强自正交性（SSO）条件来界定。本文研究一种“逆”场景：给定任意$\mathbb{F}_d$线性函数（即指定任意矩阵$V_k\in\mathbb{F}_d^{m\times m_k}$），目标是刻画所有可行通信成本元组$(\Delta_1,\cdots,\Delta_K)$的集合——不仅基于$N$-和箱协议，而且涵盖所有可能的量子编码方案。作为主要成果，我们完全解决了$K=3$个发送方的情形（$K\geq 4$的情形仍有待研究）。基于$N$-和箱协议的编码方案（辅以将量子比特视为经典比特、时分复用和批处理等基本思想）在所有情况下均被证明是信息论最优的。例如，在对称情形下：设rk$(V_1)$=rk$(V_2)$=rk$(V_3) \triangleq r_1$，rk$([V_1, V_2])$=rk$([V_2, V_3])$=rk$([V_3, V_1])\triangleq r_2$，且rk$([V_1, V_2, V_3])\triangleq r_3$（rk表示秩），则最小总下载成本为$\max \{1.5r_1 + 0.75(r_3 - r_2), r_3\}$。