Entanglement of formation

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

For a mixed bipartite state ${\displaystyle \rho _{AB}}$, 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, ${\displaystyle E_{f}(\rho _{AB})=\min \sum _{i}p_{i}E_{f}(|\psi _{i}\rangle _{AB})}$, where ${\displaystyle \rho =\sum _{i}p_{i}|\psi _{i}\rangle \langle \psi _{i}|_{AB}}$, and the minimization is over all the possible ways in which one can decompose ${\displaystyle \rho }$ 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.

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

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

where ${\displaystyle h(x)}$ is the Shannon entropy function,

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