Full Network nonlocality sharing in extended bilocal scenario via Weak Measurements with the optimal pointer

Quantum networks, which can exceed the framework of the standard Bell theorem, flourish the investigation of quantum nonlocality further. Recently, a concept of full quantum network nonlocality (FNN) which is stronger than network nonlocality (NN), has been defined and can be witnessed by Kerstjens-Gisin-Tavakoli (KGT) inequalities [Phys. Rev. Lett. 128, 010403 (2022)]. In this letter, we explored the recycling of FNN as quantum resources by analyzing the FNN sharing between different combinations of observers. The FNN sharing in an extended bilocal scenario (consisting of two independent 2-qubit quantum states as sources) via weak measurements has been completely discussed. According to the different motivations of the observer-Charlie1, two types of possible FNN sharing, passive FNN sharing and active FNN sharing, can be investigated by checking the simultaneous violation of KGT inequalities between Alice-Bob-Charlie1 and Alice-Bob-Charlie2. Our results show that passive FNN sharing is impossible while active FNN sharing can be achieved by proper measurements, which indicates that FNN sharing in this scenario requires more cooperation by the intermediate observers compared with Bell nonlocality sharing and network nonlocality sharing.sharing.