Quantifying tripartite entanglement with entropic correlations

We show how to quantify tripartite entanglement using entropies derived from experimental correlations. We use a multipartite generalization of the entanglement of formation that is greater than zero if and only if the state is genuinely multipartite entangled. We develop an entropic witness for tripartite entanglement, and we show that the degree of violation of this witness places a lower limit on the tripartite entanglement of formation. We test our results in the three-qubit regime using the GHZ-Werner state and the W-Werner state, and in the high-dimensional pure-state regime using the triple-Gaussian wavefunction describing the spatial and energy-time entanglement in photon triplets generated in third-order spontaneous parametric down-conversion. In addition, we discuss the challenges in quantifying the entanglement for progressively larger numbers of parties, and give both entropic and target-state-based witnesses of multipartite entanglement that circumvent this issue.