Figure 5

A comparison of attainable variance \(M(\hat{\rho })\) as a function of success probability. The variance is normalized with respect to the minimal variance achievable by Gaussian states. We use the same vertical and horizontal axes in the plots to show the contrast between the almost ideal (a) and lossy (b, c) scenarios with \(99\%\), \(90\%\) and \(80\%\) transmission efficiencies. Horizontal dashed lines are used to mark the optimal cubic state approximations \(\left| v^{\bigstar }\right\rangle \in \mathscr {H}_{v}\) constructed on low-dimensional TFS. We encode the information about the POVM elements as follows: APD click with solid black line, PNRD projection onto \(\left| 3\right\rangle\) with solid red, APD cascades comprising four (\({\hat{\Pi }}_{3}^{4}\), dashed magenta), five (\({\hat{\Pi }}_{3}^{5}\), magenta) and ten (\({\hat{\Pi }}_{3}^{10}\), blue) detectors where three detectors click. Overall, utilizing the PNRD \(\left| 3\right\rangle\) (solid red) produces states with lowest non-linear variance, therefore producing comparatively better approximations of the cubic state. In both (b) and (c) a single APD outperforms the APD cascades comprising five and four detectors for probabilities greater than \(1\%\). In this regime the cascade comprising ten detectors still offers advantage over single APD. In (c) a single APD outperforms APD cascades comprising either four, five or ten detectors for success probabilities larger than roughly \(5\%\).