  The certification of entanglement dimensionality is of great importance in
characterizing quantum systems. Recently, it is pointed out that quantum
correlation of high-dimensional states can be simulated with a sequence of
lower-dimensional states. Such problem may render existing characterization
protocols unreliable---the observed entanglement may not be a truly
high-dimensional one. Here, we introduce the notion of irreducible entanglement
to capture its dimensionality that is indecomposable in terms of a sequence of
lower-dimensional entangled systems. We prove this new feature can be detected
in a measurement-device-independent manner with an entanglement witness
protocol. To demonstrate the practicability of this technique, we
experimentally apply it on a 3-dimensional bipartite state and the result
certifies the existence of irreducible (at least) 3-dimensional entanglement.
