  Quantifying entanglement in a quantum system generally requires a complete
quantum tomography followed by the NP-hard computation of an entanglement
monotone --- requirements that rapidly become intractable at higher dimensions.
Observing entanglement in large quantum systems has consequently been relegated
to witnesses that only verify its existence. In this article, we show that the
violation of recent entropic witnesses of the Einstein-Podolsky-Rosen paradox
also provides tight lower bounds to multiple entanglement measures, such as the
entanglement of formation and the distillable entanglement, among others. Our
approach only requires the measurement of correlations between two pairs of
complementary observables---not a tomography---so it scales efficiently at high
dimension. Despite this, our technique captures almost all the entanglement in
common high-dimensional quantum systems, such as spatially or temporally
entangled photons from parametric down-conversion.
