Hierarchies of multipartite entanglement for continuous-variable states

We derive a hierarchy of separability criteria for multimode continuous-variable systems. They permit one to study in a unified way the $k$-partite entanglement of broad classes of Gaussian and non-Gaussian states. With specific examples we demonstrate the strength of the criteria, and we discuss their assessment based on data obtained from Gaussian measurements.

Figure 1

Density map of the inseparability properties of the Werner-type GHZ state defined in Eq. (25) in terms of the mixing $g$ and squeezing parameter $r$. The black solid line depicts the border between bipartite entangled (blue region) and separable states according to the PPT criterion. Within the former, the blue dashed and orange dot-dashed lines delimit the region of the states for which the hierarchies ${\mathcal{T}}_{2,3}$ and ${\mathcal{T}}_{3,3}$ return positive values, respectively.

Figure 2

${\mathcal{T}}_{2,2}\left(\widehat{\rho }\right)$ as a function of the amplitude $\left|\alpha \right|$ for the CPS-TSVS defined in Eq. (27) with $r=0,\alpha =\left|\alpha \right|e^{i\frac{\sqrt{2}}{2}}$, and $\beta =\left|\beta \right|e^{i\frac{\pi }{2}}$.

Figure 3

Time evolution of ${\mathcal{T}}_{2,2}(\widehat{\rho }\left(t\right))$ when the system is initially in the CPS-TSVS state plotted in Fig. 2 with $\left|\alpha \right|=0.5$, and it is in contact with independent thermal reservoirs with $N_{\mathrm{th}}=2$ (black solid line) and with $N_{\mathrm{th}}=4$ (red dashed line).