We develop an approach of quantifying entanglement in mixed quantum states bythe optimal entanglement witness operator. We identify the convex set of mixedstates for which a single witness provides the exact value of an entanglementmeasure, and show that the convexity, properties, and symmetries ofentanglement or of a target state considerably fix the form of the optimalwitness. This greatly reduces difficulty in computing and experimentallydetermining entanglement measures. As an example, we show how to experimentallyquantify bound entanglement in four-qubit noisy Smolin states and three-qubitGreenberger-Horne-Zeilinger (GHZ) entanglement under white noise. For generalmeasures and states, we provide a numerical method to efficiently optimizewitness.
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