Separability and distillability in composite quantum systems - a primer

Quantum mechanics is already 100 years old, but remains alive and full of c hallenging open problems. On one hand, the problems encountered at the fron tiers of modern theoretical physics like quantum gravity, string theories, etc. concern quantum theory, and are at the same time related to open probl ems of modern mathematics. But even within non-relativistic quantum mechani cs itself there are fundamental unresolved problems that can be formulated in elementary terms. These problems are also related to challenging open qu estions of modern mathematics; linear algebra and functional analysis in pa rticular. Two of these problems will be discussed in this article: (a) the separability problem, i.e. the question when the state of a composite quant um system does not contain any quantum correlations or entanglement; and (b ) the distillability problem, i.e. the question when the state of a composi te quantum system can be transformed to an entangled pure state using local operations (local refers here to component subsystems of a given system). Although many results concerning the above mentioned problems have been obt ained (in particular in the last few years in the framework of quantum info rmation theory), both problems remain until now essentially open. We will p resent a primer on the current state of knowledge concerning these problems , and discuss the relation of these problems to one of the most challenging questions of linear algebra: the classification and characterization of po sitive operator maps.