Fig. 1: Quantum correlations and the complexity of tomography.

Gaussian-entanglable (GE) state are states generated by performing Gaussian protocols on (possibly non-Gaussian) separable input states. The class of GE states includes the output state of the Hong–Ou–Mandel experiment, i.e., two photons input to a beam-splitter where photon bunching effect emerges, and the output states of the boson sampling protocol, where single photons together with vacuum states are input into a multi-mode linear interferometer. Other examples include entangled cat states, obtained by applying a beam-splitter to two identical cat states, and multi-mode GKP states, generated by performing Gaussian unitaries on single-mode GKP states. In addition, the class of GE states encompasses both the convex hull of Gaussian states and the entire set of separable states. Here, we prove that the NOON state \(| N\left.\right\rangle | 0\left.\right\rangle+| 0\left.\right\rangle | N\left.\right\rangle\) with N ≥ 3, superposition of TMSV states \(| {\zeta }_{r,0}\left.\right\rangle+| {\zeta }_{r,\pi }\left.\right\rangle\), and the two-mode arithmetic progression state (see Supplemental Note 4) do not belong to the set of GE states. These states are conventionally generated by applying controlled gates followed by post-selection. At last, we show that the sample complexity in learning pure GE states scales as  ~ poly(m) regarding number of modes m; while NGE state generally requires an exponential overhead \(\sim \exp (m)\).