# Entanglement of formation

The entanglement of formation is an entanglement measure for bipartite quantum states.

It is defined as

Ef(ρ) = minEpiEE(∣ψi⟩) where the minimization is over all ensembles of pure states E = {(pi, ∣ψi⟩)} that realizes the given state, ρ = ∑ipiψi⟩⟨ψi∣, and EE(∣ψ⟩) is the entropy of entanglement which is defined for pure states. This kind of extension of a quantity defined on pure states to mixed states is called a convex roof construction.

Entanglement of formation quantifies how many bell states are needed per copy of to prepare many copies of ρ using the following specific LOCC procedure:

• For each copy, select which pure state ∣ϕi⟩ to prepare from a probability distribution qi.
• For each of the different ∣ϕi⟩, prepare the required number of copies from bell states.
• Discard the information about which copy is in which pure state.

Until recently, it was not known if entanglement of formation was equal to the entanglement cost. It was shown in 2008 that the quantities are not equal, and furthermore that entanglement of formation is not additive (see section XIV D). However, the entanglement cost is equal to the regularization of the entanglement of formation,

$$E_c(\rho) = \lim_{n \to \infty} \frac{1}{n} E_f(\rho^{\otimes n}).$$