  Entanglement in high-dimensional many-body systems plays an increasingly
vital role in the foundations and applications of quantum physics. In the
present paper, we introduce a theoretical concept which allows to categorize
multipartite states by the number of degrees of freedom being entangled. In
this regard, we derive computable and experimentally friendly criteria for
arbitrary multipartite qudit systems that enable to examine in how many degrees
of freedom a mixed state is genuine multipartite entangled.
