Detecting Entanglement of Unknown Continuous Variable States with Random Measurements

We develop a scheme for the detection of entanglement in any continuous variable system, by constructing an optimal entanglement witness from random homodyne measurements. To this end, we introduce a set of linear constraints that guarantee the necessary properties of a witness and allow for its optimisation via a semidefinite program. We test our method on the class of squeezed vacuum states and study the efficiency of entanglement detection in general unknown covariance matrices. The results show that we can detect entanglement, including bound entanglement, in arbitrary continuous variable states with fewer measurements than in full tomography. The statistical analysis of our method shows a good robustness to statistical errors in experiments.