Certifying emergent genuine multipartite entanglement with a partially blind witness

Genuine multipartite entanglement underlies correlation experiments corroborating quantum mechanics and it is an expedient empowering many quantum technologies. One of many counterintuitive facets of genuine multipartite entanglement is its ability to exhibit an emergent character. That is, one can infer its presence in some multipartite states merely from a set of its separable marginals. Here we show that the effect can also be found in the context of Gaussian states of bosonic systems. Specifically, we construct examples of multimode Gaussian states carrying genuine multipartite entanglement which can be verified solely from separable nearest-neighbor two-mode marginals. The key tool of our construction is an entanglement witness acting only on some two-mode reductions of the global covariance matrix, which we find by a numerical solution of a semidefinite program. We also propose an experimental scheme for preparation of the simplest three-mode state, which requires interference of three correlatively displaced squeezed beams on two beam splitters. Besides revealing the concept of emergent genuine multipartite entanglement in the Gaussian scenario and bringing it closer to experimentally testable form, our results pave the way to effective diagnostics methods of global properties of multipartite states without complete tomography.

Figure 1

Graphical representation of all minimal sets of two-mode marginals for three and four modes. The vertices represent the modes. An edge connecting modes $i$ and $j$ represents a two-mode marginal density matrix ${\rho }_{ij}$ and it is labeled by the respective covariance matrix ${\gamma }_{ij}$. See text for details.

Figure 2

Decomposition of symplectic transformation $S$ generating a Gaussian state with CM ${\gamma }_3$ of three modes $A,B$, and $C$: ${\nu }_j$ thermal states with mean number of thermal photons $({\nu }_j-1)/2$, $j=A,B,C$ (circles); $U$: passive transformation consisting of beam splitters ${\mathrm{BS}}_{jk}^{\left(U\right)}$, $jk=AB,AC,BC$ (leftmost box); $V$: passive transformation consisting of beam splitters ${\mathrm{BS}}_{jk}^{\left(V\right)}$ (rightmost box); $R$ squeezing transformation consisting of one squeezer in position quadrature, $R_A$, and two squeezers in momentum quadrature, $R_B$ and $R_C$ (middle box). For rounded parameters as in Tables 5 and 6, the circuit produces the CM ${\gamma }_3^{\prime }$, which closely approximates the CM ${\gamma }_3$, and retains its entanglement properties. See text for details.

Figure 3

Scheme for preparation of a Gaussian state with CM ${\overline{\gamma }}_3$ carrying genuine multipartite entanglement verifiable from nearest-neighbor separable marginals. The input comprises three vacuum states (circles). The squeezing transformation $R$ (leftmost box) and the transformation $V$ (rightmost box) are the same as in Fig. 2. The block $D$ (middle box) contains correlated displacements $D_A,D_B$, and $D_C$ (white boxes) given in Eq. (27), where the parameters ${\alpha }_j$ and ${\beta }_j$ are in Table 8 and $\langle t^2\rangle =\langle w^2\rangle =({\nu }_A-1)/2$. See text for details.