Fig. 2: The GDE tomography protocol simply using local state tomography.

First, we estimate the displacement α and covariance matrix V by homodyne and heterodyne measurements. Based on the results, we subsequently apply a calibrated Gaussian unitary \({\hat{U}}_{\widetilde{S}}^{{{\dagger}} }{\hat{U}}_{\widetilde{{{{\boldsymbol{\alpha }}}}}}^{{{\dagger}} }\) to counter-rotate the state, and finally perform local tomography to reconstruct the local states. When the symplectic eigenvalues are non-degenerate, the Gaussian entangling unitary \({\hat{U}}^{g}\) is fully identified and the counter-rotated states is separable; Otherwise, the state becomes passive-separable and we continue with the process depicted in Fig. 3.