Quantifying mixed-state quantum entanglement by optimal entanglement witnesses

We develop an approach of quantifying entanglement in mixed quantum states by the optimal entanglement witness operator. We identify the convex set of mixed states for which a single witness provides the exact value of an entanglement measure and show that the convexity, properties, and symmetries of entanglement or of a target state considerably fix the form of the optimal witness. This greatly reduces the difficulty in computing and experimentally determining entanglement measures. As an example, we show how to experimentally quantify bound entanglement in four-qubit noisy Smolin states and three-qubit Greenberger-Horne-Zeilinger entanglement under white noise. For general measures and states, we provide a numerical method to efficiently optimize the witness.