  The emergence of quantum devices has raised a significant issue: how to
certify the quantum properties of a device without placing trust in it. To
characterise quantum states and measurements in a device-independent way, up to
some degree of freedom, we can make use of a technique known as self-testing.
While schemes have been proposed to self-test all pure multipartite entangled
states (up to complex conjugation) and real local rank-one projective
measurements, little has been done to certify mixed entangled states, composite
or non-projective measurements. By employing the framework of quantum networks,
we propose a scheme for self-testing (up to complex conjugation) arbitrary
extremal measurements, including the projective ones, but also in an indirect
way any quantum states, including the mixed ones. The quantum network
considered in this work is the simple star network, which is implementable
using current technologies. For our purposes, we also construct a scheme that
can be used to self-test the two-dimensional tomographically complete set of
measurements with an arbitrary number of parties.
