Fig. 4: Performance of Gaussian-disentangling protocol (GDE) versus direct tomography (RρR).

Here, we numerically simulate the quantum state tomography process on a two-mode GE state with the true state being number state \(| 0\left.\right\rangle \otimes | 1\left.\right\rangle\), cat state \({| {{{\rm{cat}}}}\left.\right\rangle }_{0.1}\otimes {| {{{\rm{cat}}}}\left.\right\rangle }_{1.1}\) and thermal state ρ_th,n=0.2 ⊗ ρ_th,n=0.3 correlated by a beam-splitter (with a rotation angle θ = π/4). The error, defined as the trace distance between the true state and the reconstructed state obtained through direct tomography using the standard RρR algorithm², is shown as a function of the number of copies M used in tomography (black squares for ideal case and grey stars for the lossy case). In the lossy case, pure loss channel (with a transmissivity ratio η = 0.9) is applied to both modes after the beam splitter. In contrast, the bottom envelope of achievable errors in the proposed Gaussian disentangling (GDE) algorithm, after minimising over possible sample number in estimating displacements and the covariance matrix, is given by red circles and pink triangles (lossy case). Their linear fit shows errors in GDE protocol decrease faster (with slope k) compared to RρR method in all the three different states, demonstrating the advantage of GDE method.