  In the past decades, quantum entanglement has been recognized to be the basic
resource in quantum information theory. A fundamental need is then the
understanding its qualification and its quantification: Is the quantum state
entangled, and if it is, then how much entanglement is carried by that? These
questions introduce the topics of separability criteria and entanglement
measures, both of which are based on the issue of classification of
multipartite entanglement. In this dissertation, after reviewing these three
fundamental topics for finite dimensional Hilbert spaces, I present my
contribution to knowledge. My main result is the elaboration of the partial
separability classification of mixed states of quantum systems composed of
arbitrary number of subsystems of Hilbert spaces of arbitrary dimensions. This
problem is simple for pure states, however, for mixed states it has not been
considered in full detail yet. I give not only the classification but also
necessary and sufficient criteria for the classes, which make it possible to
determine to which class a mixed state belongs. Moreover, these criteria are
given by the vanishing of quantities measuring entanglement. Apart from these,
I present some side results related to the entanglement of mixed states. These
results are obtained in the learning phase of my studies and give some
illustrations and examples.
