Convexification of the Quantum Network Utility Maximisation Problem

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.