Entanglement of formation and concurrence

In both classical mechanics and quantum mechanics, one can de ne a pure state to be a state that is as completely speci ed as the theory allows. In classical mechanics a pure state might be represented by a point in phase space. In quantum mechanics it is a vector in a complex vector space. Perhaps the most remarkable feature of quantum mechanics, a feature that clearly distinguishes it from classical physics, is this: for any composite system, there exist pure states of the system in which the parts of the system do not have pure states of their own. Such states are called entangled. One can also de ne the concept of entanglement for mixed quantum states: a mixed state is entangled if it cannot be represented as a mixture of unentangled pure states. For both pure and mixed quantum states, there are good measures of the degree of entanglement. In the case of pure states of a bipartite system there is a single widely accepted measure of entanglement, whereas for mixed states of such systems there are three measures that have been extensively studied. One of these, entanglement of formation, is the subject of this paper. After recalling the de nition of entanglement of formation and the motivation for the de nition, we review some aspects of the current state of our understanding of this entanglement measure. For the special case of a pair of qubits, the entanglement of formation is closely related to a simpler but less physically motivated measure of entanglement called the concurrence, and we devote much of our discussion to this concept and to possible generalizations of it to other bipartite systems.We begin, though, by discussing the simplest case, the entanglement of pure states.