Network Utility Maximisation (NUM) addresses the problem of allocating resources fairly within a network and explores the ways to achieve optimal allocation in real-world networks. Although extensively studied in classical networks, NUM is an emerging area of research in the context of quantum networks. In this work, we consider the quantum network utility maximisation (QNUM) problem in a static setting, where a user's utility takes into account the assigned quantum quality (fidelity) via a generic entanglement measure as well as the corresponding rate of entanglement generation. Under certain assumptions, we demonstrate that the QNUM problem can be formulated as an optimisation problem with the rate allocation vector as the only decision variable. Using a change of variable technique known in the field of geometric programming, we then establish sufficient conditions under which this formulation can be reduced to a convex problem, a class of optimisation problems that can be solved efficiently and with certainty even in high dimensions. We further show that this technique preserves convexity, enabling us to formulate convex QNUM problems in networks where some routes have certain entanglement measures that do not readily admit convex formulation, while others do. This allows us to compute the optimal resource allocation in networks where heterogeneous applications run over different routes.

翻译：网络效用最大化（NUM）旨在解决网络中资源的公平分配问题，并探索在实际网络中实现最优分配的途径。尽管在经典网络中已得到广泛研究，NUM在量子网络背景下仍是一个新兴的研究领域。本文考虑静态场景下的量子网络效用最大化问题，其中用户效用通过通用的纠缠度量来考量所分配的量子质量（保真度）以及相应的纠缠生成速率。在一定假设下，我们证明QNUM问题可表述为以速率分配向量为唯一决策变量的优化问题。借助几何规划领域中已知的变量替换技术，我们进一步建立了将该表述简化为凸问题的充分条件——凸优化问题即使在较高维度下也能被高效且确定性地求解。我们进一步证明该技术能保持凸性，使得我们能够在某些路径的纠缠度量不易直接形成凸表述、而其他路径可以的情况下，构建凸的QNUM问题。这使我们能够计算在异构应用程序运行于不同路径的网络中的最优资源分配。