Quantitative entanglement witnesses

TLDR

Entanglement witnesses enable detection of entanglement without full state tomography, and a negative expectation value directly signals entanglement, including multipartite entanglement. This paper shows that such tests, based on the same data, can provide quantitative estimates of entanglement. When a test is strongly violated, the state can be inferred to be quantitatively highly entangled. The authors analyze several entanglement measures—negativity, entanglement of formation, and robustness—in bipartite and multipartite settings, illustrating their use with recent quantum state‑preparation experiments.

Abstract

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 paper, 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.