Quantifying high-dimensional entanglement with Einstein-Podolsky-Rosen correlations

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 $\textit{witnesses}$ that only verify its existence. In this Letter, 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 $\textit{entanglement of formation}$ and the $\textit{distillable entanglement}$, 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.

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