Entanglement criterion for four-partite systems based on local sum uncertainty relations

We propose an experimentally feasible criterion for the detection of entanglement in an arbitrary four-partite quantum system. It is written as local sum uncertainty relations for suitably chosen observables of the subsystems. The relations are satisfied by all separable states so that their violation implies the presence of the entanglement. This criterion generalizes the tighter criterion for tripartite systems introduced in our previous work (Y. Akbari-Kourbolagh and M. Azhdargalam, Phys. Rev. A 97, 2018) to the four-partite cases. It can be used to detect the entanglement of both discrete- and continuous-variable quantum states without having to know their complete density matrices. Moreover, in contrast to the well-established positive partial transpose criterion, it is also able to detect bound entangled states. Its efficiency is shown by the examples of four-qubit and Gaussian four-mode states.