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 $|\psi \rangle _{AB}$, using Schmidt decomposition, we see that the reduced density matrices of A and B have the same form $\rho _{A}=\rho _{B}$. The von Neumann entropy $S(\rho _{A})=S(\rho _{B})$ of the reduced density matrix can be used to measure the entanglement of the state $|\psi \rangle _{AB}$. We denote this kind of measure as $E_{f}(|\psi \rangle _{AB})=S(\rho _{A})=S(\rho _{B})$, and called it the entanglement entropy. This is also known as the entanglement of formation of pure state.

For a mixed bipartite state $\rho _{AB}$, a natural generalization is to consider all the ensemble realizations of the mixed state. We can define a quantity by minimizing over all these ensemble realizations, $E_{f}(\rho _{AB})=\min \sum _{i}p_{i}E_{f}(|\psi _{i}\rangle _{AB})$, where $\rho =\sum _{i}p_{i}|\psi _{i}\rangle \langle \psi _{i}|_{AB}$, and the minimization is over all the possible ways in which one can decompose $\rho$ as a mixture of pure states. This kind of extension of a quantity defined on pure states to mixed states is called a convex roof construction. This quantity is called the entanglement of formation.

Relation with concurrence

The entanglement of formation has a close relationship with concurrence. For a given state $\rho _{AB}$, its entanglement of formation $E_{f}(\rho _{AB})$ is related to its concurrence $C$:

$E_{f}=h\left({\frac {1+{\sqrt {1-C^{2}}}}{2}}\right)$

where $h(x)$ is the Shannon entropy function,

$h(x)=-x\log _{2}x-(1-x)\log _{2}(1-x).$