  Entanglement of high-dimensional quantum systems has become increasingly
important for quantum communication and experimental tests of nonlocality.
However, many effects of high-dimensional entanglement can be simulated by
using multiple copies of low-dimensional systems. We present a general theory
to characterize those high-dimensional quantum states for which the
correlations cannot simply be simulated by low-dimensional systems. Our
approach leads to general criteria for detecting multilevel entanglement in
multiparticle quantum states, which can be used to verify these phenomena
experimentally.
