  Quantum entanglement is the key resource for quantum information processing.
Device-independent certification of entangled states is a long standing open
question, which arouses the concept of self-testing. The central aim of
self-testing is to certify the state and measurements of quantum systems
without any knowledge of their inner workings, even when the used devices
cannot be trusted. Specifically, utilizing Bell's theorem, it is possible to
place a boundary on the singlet fidelity of entangled qubits. Here, beyond this
rough estimation, we experimentally demonstrate a complete self-testing process
for various pure bipartite entangled states up to four dimensions, by simply
inspecting the correlations of the measurement outcomes. We show that this
self-testing process can certify the exact form of entangled states with
fidelities higher than 99.9% for all the investigated scenarios, which
indicates the superior completeness and robustness of this method. Our work
promotes self-testing as a practical tool for developing quantum techniques.
