  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 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 (GHZ) entanglement under white noise. For general
measures and states, we provide a numerical method to efficiently optimize
witness.
