Unbounded sharing of network nonlocality in the two-forked tree-shaped scenario

Quantum network nonlocality plays a crucial role in long-distance quantum communication, and nonlocality sharing via sequential measurements provides a novel perspective for large-scale quantum network applications. In this paper, we investigate network nonlocality sharing in the two-forked tree-shaped network. For the twoinput n-layer (n 2) scenario, the number of sequential observers in existing schemes is constrained by the layer n and the number of sequential sides. Here we achieve unbounded sharing of network nonlocality for arbitrary n by introducing sharpness parameters into Bell state measurements and allowing sequential observers to perform measurements with different sharpness parameters. An unbounded number of independent observers on each of the m (m = 1,. .., 2n−2 ) sequential sides can simultaneously share non-(2n − 2)-locality by violating the two-input (2n − 2)-local inequality with all other nonsequential observers. Furthermore, we extend the twoinput scenario to a generalized d-input (d 2) case, investigating how the input number d affects network nonlocality sharing. A general model for sequential measurement scenarios in d-input two-forked n-layer treeshaped networks is established. Using the simplest case (n = 3) as an example, we investigate non-six-locality sharing based on the d-input six-local inequality and establish the relationship between the number of sequential observers, the number of sequential sides, and the input d. Based on the results, we conjecture that the two-input configuration is optimalfor nonlocality sharing in two-forked tree-shaped networks.