Entanglement of formation
The entanglement of formation is a quantity that measures the entanglement of a bipartite quantum state. Definition
For a pure bipartite quantum state , using Schmidt decomposition, we see that the reduced density matrices of A and B have the same form . The von Neumann entropy  of the reduced density matrix can be used to measure the entanglement of the state . We denote this kind of measure as , and called it the entanglement entropy. This is also known as the entanglement of formation of pure state. For a mixed bipartite state , a natural generalization is to consider all the ensemble realizations of the mixed state. We define the entanglement of formation more mixed states by minimizing over all these ensemble realizations,
, where the infimum is taken over all the possible ways in which one can decompose  into pure states . This kind of extension of a quantity defined on some set (here the pure states) to its convex hull (here the mixed states) is called a convex roof construction. Properties 
Entanglement of formation is quantifying how much entanglement (measured in ebits) is necessary on average to prepare the state. The measure clearly coincides with entanglement entropy for pure states. It is zero for all separable states and non-zero for all entangled states. By construction  is convex. Entanglement of formation is known to a non-additive measure of entanglement. That is, there are bipartite quantum states  such that the entanglement of formation of the joint state  is smaller than the sum of the individual states' entanglement, i. e., . Note that for other states (for example pure or separable states) equality holds. Furthermore, it has been shown that the regularized entanglement of formation equals the entanglement cost. That is, for large  the entanglement of formation of  copies of a state  divided by  converges to the entanglement cost

The non-additivity of  thus implies that there are quantum states for which there is a “bulk discount” when preparing them from pure states by local operations: it is cheaper, on average, to prepare many together than each one separately. Relation with concurrence
For states of two qubits, the entanglement of formation has a close relationship with concurrence. For a given state , its entanglement of formation  is related to its concurrence :

where  is the Shannon entropy function,

References

Quantum information theory