Entanglement quantification from incomplete measurements: applications using photon-number-resolving weak homodyne detectors

The certificate of success for a number of important quantum information processing protocols, such as entanglement distillation, is based on the difference in the entanglement content of the system quantum states before and after the protocol. In such cases, effective bounds need to be placed on the entanglement of non-local states consistent with statistics obtained from local measurements. In this paper, we study numerically the ability of a hybrid homodyne detector that combines phase sensitivity and photon-number resolution to set accurate bounds on the entanglement content of two-mode quadrature squeezed states without the need for full state tomography. We show that it is possible to set tight lower bounds on the entanglement of a family of two-mode degaussified states using only a few measurements. This presents a significant improvement over the resource requirements for the experimental demonstration of continuous-variable entanglement distillation, which traditionally relies on full quantum state tomography.