Entanglement of formation

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

It is defined as

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where the minimization is over all ensembles of pure states \mathcal{E} = \{(p_i,|\psi_i\rangle)\} that realizes the given state, \rho = \sum_i p_i |\psi_i \rangle \langle \psi_i|, and E_E(|\psi \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 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 |\phi_i\rangle to prepare from a probability distribution qi.
  • For each of the different |\phi_i\rangle, prepare the required number of copies from bell states.
  • Discard the information about which copy is in which pure state.

It is not known if the entanglement of formation is equal to the entanglement cost in general. However, the entanglement cost is equal to the regularization of the entanglement of formation,

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