Detecting non-Gaussian entanglement beyond Gaussian criteria

Abstract

Entanglement is central to quantum theory, yet detecting it reliably in non-Gaussian systems remains a long-standing challenge. In continuous-variable platforms, inseparability tests based on Gaussian statistics - such as those of Duan and Simon - fail when quantum correlations are encoded in higher moments of the field quadratures. Here we introduce an inseparability criterion that exposes non-Gaussian entanglement that escapes covariance-based criteria by incorporating higher-order quadrature cumulants. The criterion extends Gaussian theory without requiring full state tomography and can be evaluated directly from homodyne and heterodyne data and is possible to extend to arbitrary superpositions of Fock states in two modes. This provides an experimentally viable approach for identifying non-Gaussian resources in continuous-variable platforms.

Figures (5)

• Figure 1: The inseparability of the split lossy single photon vs. the transmittivity of the pure loss channel. The Wigner function of the single photon is approximated by four Gaussians in phase space, and the inseparability criterion is plotted for two different fidelities of the approximation.
• Figure 2: The inseparability criterion for the split lossy PhSSV state $\hat{\rho}_1^\eta$ vs. the transmittivity of the pure loss channel for two values of squeezing (solid lines). The Wigner logarithmic negativity (WLN) for the $r=10^{-3}$ state is computed for a small number of points (triangles).
• Figure 3: Preparation of the PhSSV state. The transmittivity of the first beam-splitter $\hat{B}_{01}(\theta)$ is set to $\theta=\arccos(\sqrt{0.99})$. The PhSSV state passes through a pure loss channel with transmittivity $\eta$, after which the state is entangled with vacuum in mode 2 via the second beam-splitter $\hat{B}_{12}(\frac{\pi}{4})$.
• Figure 4: Entanglement Witness: Shown here are calculated values of the violation of the criterion quantified by (RHS-LHS) for a Photon subtracted squeezed vacuum (PhSSV) split on a beamsplitter as well as a Squeezed vacuum split on a beam splitter. In the case of the Gaussian state we see the expected entanglement as the separability condition is violated for all values $r>0$ for both criteria. However, for the non-Gaussian state there is an increased violation of the inequality for the criterion derived here. This may be attributed to the increasing value of the $4^{\text{th}}$ order moments of the EPR type operators as well as the appearance of the $2,2$-joint cumulants. Furthermore the Duan criterion misdiagnoses the non-Gaussian entangled state as separable until a threshold squeezing of approximately $r=0.55$.
• Figure 5: For a two mode squeezed vacuum state and the split squeezed vacuum the criterion is tightly violated for all $r>0$ while separability is achieved for the limiting case of no squeezing as the criterion collapses to that of a product of vacuum states. Furthermore, note that the violation is increased for the TMSV as compared to a split squeezed vacuum as one would expect considering stronger correlations in the TMSV state as compared to the the split squeezed vacuum.