Hierarchy of quantum operations in manipulating coherence and entanglement

Quantum resource theory under different classes of quantum operations advances multiperspective understandings inherent quantum-mechanical properties, such as coherence and entanglement. We establish hierarchies for manipulating entanglement in distributed settings, where at least one the two spatially separated parties are restricted from generating coherence. In these we introduce new also characterize those maximal, i.e., resource-non-generating operations, progressing beyond existing studies on incoherent versions local classical communication separable operations. The maximal admit a semidefinite-programming formulation useful numerical algorithms, whereas not. To hierarchies, prove sequence inclusion relations among by clarifying tasks separation appears. demonstrate an asymptotically non-surviving hierarchy terms performance task assisted distillation, one-shot scenario vanishes asymptotic limit. Our results serve fundamental analytical tools to investigate interplay between theory.