Semi-device-independent characterization of multipartite entanglement of states and measurements

The semi-device-independent framework allows one to draw conclusions about properties of an unknown quantum system under weak assumptions. Here we present a semi-device-independent scheme for the characterization of multipartite entanglement based around a game played by several isolated parties whose devices are uncharacterized beyond an assumption about the dimension of their Hilbert spaces. Our scheme can simultaneously certify that an $n$-partite high-dimensional quantum state features genuine multipartite entanglement, and that a joint measurement on $n$ subsystems is entangled. Moreover, it provides a lower bound on the number of entangled measurement operators. These tests are strongly robust to noise, and even optimal for certain classes of states and measurements, as we demonstrate with illustrative examples. Notably, our scheme allows the certification of many entangled states admitting a local model, which therefore cannot violate any Bell inequality.