The Shape of Information: Global Information Geometric Limits in Multi-task Quantum Systems
Future quantum networks are expected to perform multiple tasks simultaneously within a single system, such as integrated sensing and communication (ISAC) architectures. Despite various metrics, we find that the evaluation of multiple tasks can be unified by their information capacity, and the total task capacity is not determined simply by addition, but is fundamentally constrained by an information geometry which we call the global quantum Fisher information matrix (g-QFIM). With this insight, we derive a non-asymptotic, measurement-independent upper bound on the Holevo information for multi-task systems, which takes a Shannon-capacity-like form. It not only quantifies the capacity limit, but also the allocability. Our results reveal a structural phase transition in multi-task performance under resource variation, where additional physical resources no longer increase independent task capacity but instead concentrate more information into a new mono-task mode. Numerical simulations based on photonic phase encoding and realistic noise channels confirm these predictions. This work establishes a unified information-geometric principle for quantum multi-task systems, with implications for the design of future quantum networks and ISAC architectures.