Entanglement of formation

The '''entanglement of formation''' is an [[entanglement measure]] for [[bipartite]] quantum states. It is defined as :$E\_f(\backslash rho)\; =\; \backslash min\_\backslash mathcal\{E\}\; p\_i\; E\_E(|\backslash psi\_i\; \backslash rangle)$ where the minimization is over all [[ensemble]]s of pure states $\backslash mathcal\{E\}\; =\; \backslash \{(p\_i,|\backslash psi\_i\backslash rangle)\backslash \}$ that realizes the given state, $\backslash rho\; =\; \backslash sum\_i\; p\_i\; |\backslash psi\_i\; \backslash rangle\; \backslash langle\; \backslash psi\_i|$, and $E\_E(|\backslash psi\; \backslash rangle)$ 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 state]]s are needed per copy of to prepare many copies of $\backslash rho$ using the following specific [[LOCC]] procedure: * For each copy, select which pure state $|\backslash phi\_i\backslash rangle$ to prepare from a probability distribution $q\_i$. * For each of the different $|\backslash phi\_i\backslash rangle$, 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, :$E\_c(\backslash rho)\; =\; \backslash lim\_\{n\; \backslash to\; \backslash infty\}\; \backslash frac\{1\}\{n\}\; E\_f(\backslash rho^\{\backslash otimes\; n\}).$ [[Category:Entanglement]] [[Category:Mathematical structure]]