Quantitative entanglement witnesses

Entanglement witnesses provide tools to detect entanglement in experimental situations without the need of having full tomographic knowledge about the state. If one estimates in an experiment an expectation value smaller than zero, one can directly infer that the state has been entangled, or specifically multi-partite entangled, in the first place. In this article, we emphasize that all these tests - based on the very same data - give rise to quantitative estimates in terms of entanglement measures: "If a test is strongly violated, one can also infer that the state was quantitatively very much entangled". We consider various measures of entanglement, including the negativity, the entanglement of formation, and the robustness of entanglement, in the bipartite and multipartite setting. As examples, we discuss several experiments in the context of quantum state preparation that have recently been performed.