Entanglement of Formation

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The entanglement of formation is an entanglement monotone defined as

E_{F}(\rho,\{A,B\}) \equiv \min\limits_{\{p_i,|\psi\rangle_i\}}\sum_i p_i E( |\psi\rangle_i,\{A,B\})

where the \{p_i,|\psi\rangle_i\} satisfy the condition that

\rho=\sum_i p_i |\psi\rangle_i \langle\psi_i|. This quantity is difficult to compute for mixed states but reduces to the entropy of entanglement for pure states.

In the special case of a mixed state of two spin-1/2 particles, the entanglement of formation can be computed from the two-particle concurrence, \mathcal{C}(\rho). Therefore, it is generally possible to compute the entanglement of formation between two spins {i,j} removed from an N-spin state Ψ. The entanglement of formation for such a reduced system is a strong measure of the robustness of that state's entanglement to particle loss.

Explicitly, for the two particle state \rho=\mathrm{Tr}_{k\neq i, j}(|\Psi\rangle\langle\Psi|) we have

E_{F}(\rho,\{i,j\})=h(\frac{1}{2}[1+\sqrt{1-\mathcal{C}(\rho)^{2}}])

where h(x) = − xlog2(x) − (1 − x)log2(1 − x) and the concurrence is defined here.

References

1) Scott Hill and William K. Wootters, Entanglement of a Pair of Quantum Bits, 1997.
2) William K. Wootters, Entanglement of Formation of an Arbitrary State of Two Qubits 1998.

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