nonclassical states, also denoted as “entangled states”, can provide us with something else than paradoxes. They may be *used* to perform tasks that cannot be achieved with classical states. As benchmarks of this turning point in our view of such nonclassical states, one might mention the spectacular discoveries of (entanglement-based) quantum cryptography (1991) [@Ekert], quantum dense coding (1992) [@BW], and quantum teleportation (1993) [@BBCJPW].

In this chapter, we will consider both bipartite and multipartite composite systems. We will define formally what entangled states are, present some important criteria to discriminate entangled states from separable ones, and show how they can be classified according to their capability to perform some precisely defined tasks. Our knowledge in the subject of entanglement is still far from complete, although significant progress has been made in the recent years and very active research is currently underway. We will consider multipartite quantum states (states of more than two parties) in Section \[sec:multi\], until then, we consider only bipartite quantum states.

Bipartite pure states: Schmidt decomposition
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Consider a bipartite system in a shared pure state. The two parties in possession of the system are traditionally denoted as Alice (A) and Bob (B), who can be located 