Genuine Multipartite Einstein-Podolsky-Rosen Steering

We develop the concept of genuine $N$-partite Einstein-Podolsky-Rosen (EPR) steering. This nonlocality is the natural multipartite extension of the original EPR paradox. Useful properties emerge that are not guaranteed for genuine multipartite entangled states. In particular, there is a close link with the task of one-sided, device-independent quantum secret sharing. We derive inequalities to demonstrate multipartite EPR steering for Greenberger-Horne-Zeilinger and Gaussian continuous variable states in loophole-free scenarios.

Figure 1

Depiction of one of the bipartitions $\{A_s,B_s{\}}_{st}$ of the hybrid local hidden state (LHS) local-nonlocal model for $N=4$. Here three sites can share a Bell nonlocality, but four cannot. The group $A_s$ “steers” system $B_s$, if this model fails, when it is also constrained that $B_s$ be consistent with a local quantum state.

Figure 2

(a) Genuine CV tripartite entanglement for beams $\{1,2,3\}$ can be generated via squeezed states and beam splitters (BS) [17]. (b) Genuine tripartite entanglement exists for $\{1,2^{\prime },3^{\prime }\}$ when $\mathrm{En}{t}_3<1$, whereas EPR steering of $\{1\}$ by $\{2^{\prime },3^{\prime }\}$ is confirmed when ${\mathrm{EPR}}_3={\Delta }_{\inf ,A^{\prime }}{x}_1{\Delta }_{\inf ,A^{\prime }}{p}_1<1$. Here, $\eta$ is the efficiency of transmission of the eavesdropping beam splitters. Cases $N=2$, 4 are also shown. The loss of steering reveals the eavesdropper.