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The emergence of quantum devices has raised a significant issue: how tocertify the quantum properties of a device without placing trust in it. Tocharacterise quantum states and measurements in a device-independent way, up tosome 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 entangledstates (up to complex conjugation) and real local rank-one projectivemeasurements, little has been done to certify mixed entangled states, compositeor non-projective measurements. By employing the framework of quantum networks,we propose a scheme for self-testing (up to complex conjugation) arbitraryextremal measurements, including the projective ones, but also in an indirectway any quantum states, including the mixed ones. The quantum networkconsidered in this work is the simple star network, which is implementableusing current technologies. For our purposes, we also construct a scheme thatcan be used to self-test the two-dimensional tomographically complete set ofmeasurements with an arbitrary number of parties.

A universal scheme to self-test any quantum state and extremal measurement