Bilocal versus nonbilocal correlations in entanglement-swapping experiments

Entanglement swapping is a process by which two initially independent quantum systems can become entangled and generate nonlocal correlations. To characterize such correlations, we compare them to those predicted by bilocal models, where systems that are initially independent are described by uncorrelated states. We extend in this paper the analysis of bilocal correlations initiated in [Phys. Rev. Lett. 104, 170401 (2010)]. In particular, we derive new Bell-type inequalities based on the bilocality assumption in different scenarios, we study their possible quantum violations, and we analyze their resistance to experimental imperfections. The bilocality assumption, being stronger than Bell's standard local causality assumption, lowers the requirements for the demonstration of quantumness in entanglement-swapping experiments.

Figure 1

Typical entanglement swapping scenario where three parties, Alice, Bob, and Charlie, share two sources $S_1$ and $S_2$ that each emit independent pairs of particles in some quantum states ${\varrho }_1$ and ${\varrho }_2$. Bob performs a joint measurement $y$ on the two particles he receives from each source and obtains an output $b$. Depending on Bob's outcome, Alice and Charlie's systems end up in one out of different possible entangled states. Alice and Charlie apply some measurements $x$ and $z$ on their particle and obtain outputs $a$ and $c$. Such an experiment is characterized by a joint probability distribution $P(a,b,c|x,y,z)$.

Figure 2

The natural counterpart of the entanglement swapping scenario of Fig. 1 in terms of a locally causal model with two independent sources of hidden states: The systems produced by the source $S_1$ are characterized by hidden states ${\lambda }_1$, while those from the source $S_2$ are characterized by hidden states ${\lambda }_2$. The two sources are assumed to be independent, hence the joint distribution $\rho ({\lambda }_1,{\lambda }_2)$ of hidden states has the product form $\rho ({\lambda }_1,{\lambda }_2)={\rho }_1\left({\lambda }_1\right){\rho }_2\left({\lambda }_2\right)$, as in Eq. (2).

Figure 3

Projection of the tripartite correlation space in the $(I,J)$ plane, for $I=I^{22}$ or $I^{14}$ and $J=J^{22}$ or $J^{14}$, as defined in Eqs. (18, 19) or (21)–(22), for the “22” and “14” cases, respectively The nonconvex bilocal set $\mathcal{B}$ is delimited by the four portions of parabolas, corresponding to the inequality $\sqrt{\left|I\right|}+\sqrt{\left|J\right|}\le 1$ [Eqs. (20) or (23)]. It is included in the local set $\mathcal{L}$ delimited by the dashed lines, for which $\left|I\right|+\left|J\right|\le 1$. The set $\mathcal{Q}$ of quantum correlations is also limited in this plane by $\left|I\right|+\left|J\right|\le 1$. In the 22 case, the nonsignaling polytope ${\mathcal{N}S}^{22}$ is delimited by the outer dotted square, defined by $\max \left(\right|I^{22}|,|J^{22}\left|\right)\le 1$; in the 14 case, the projection of the nonsignaling polytope ${\mathcal{N}S}^{14}$ coincides with that of the local and quantum sets. The figure can also be understood as a two-dimensional slice of the correlation space, containing the quantum correlation $P_Q=P_Q^{22}$ (34) or $P_Q^{14}$ (30), the fully random correlation $P_0=P_0^{22}$ or $P_0^{14}$, and the bilocal correlations $P_I=P_I^{22}$ or $P_I^{14}$ and $P_J=P_J^{22}$ or $P_J^{14}$, as defined in footnotes 13 and 17. In this slice, the bilocal set is star-convex. One can see that the quantum correlation $P_Q=\frac{1}{2}{P}_I+\frac{1}{2}{P}_J$ is local, but not bilocal. When adding some white noise, it enters the bilocal set for visibilities $V\le \frac{1}{2}$ (see Sec. 3a1).

Figure 4

Projection of the tripartite correlation space in the $(I^{13},J^{13})$ plane, as defined in Eqs. (25) and (26), for the “13 case”. The projections of the bilocal set ${\mathcal{B}}^{13}$, of the local polytope ${\mathcal{L}}^{13}$, of the quantum set ${\mathcal{Q}}^{13}$ and of the nonsignaling polytope ${\mathcal{N}S}^{13}$ are similar to those of Fig. 3. When restricting to correlations with a random marginal ${\langle B^0\rangle }_{P^{13}}$ for Bob and random bipartite marginals ${\langle A_x{C}_z\rangle }_{P^{13}}$ for Alice-Charlie, the local (${\mathcal{L}}_{\text{RND}}^{13}$), quantum (${\mathcal{Q}}_{\text{RND}}^{13}$), and nonsignaling (${\mathcal{N}S}_{\text{RND}}^{13}$) sets are instead delimited by $|I^{13}|+2|J^{13}|\le 1$ (dashed diamond). The restriction of the figure to the dashed diamond can also be understood as a two-dimensional slice of the correlation space, containing the quantum correlation $P_Q^{13}$ (36), the random correlation $P_0^{13}$, and the bilocal correlations $P_I^{13}$ and $P_J^{13}$, as defined in footnote 18. One can see that the quantum correlation $P_Q^{13}=\frac{2}{3}{P}_I^{13}+\frac{1}{3}{P}_J^{13}$ is local, but not bilocal. When adding some noise, it enters the bilocal set for visibilities $V\le \frac{2}{3}$ (see Sec. 3b2).

Figure 5

(Non-)bilocality of correlations obtained by the strategy described in the main text, in the case where Alice and Charlie's detectors have a detection efficiency $\eta$. The correlations are bilocal for visibilities $V=v_1{v}_2\le V_{\mathrm{biloc}}^{\eta }$, with $V_{\mathrm{biloc}}^{\eta }$ depending on $\eta$. As explained in the text, $V_{\mathrm{biloc}}^{\eta }$ was estimated by matching lower bounds given by explicit bilocal decompositions (see Table ) and upper bounds obtained from convex relaxations of the bilocality constraint; our inequality (23) also gives an upper bound on $V_{\mathrm{biloc}}^{\eta }$; this bound was found to be tight only for $\eta \ge \frac{3}{4}$.

Figure 6

(Shaded area) Local ($V_{\mathrm{loc}}$) versus bilocal ($V_{\mathrm{biloc}}$) visibility thresholds for quantum correlations obtainable in standard entanglement swapping experiments, with binary inputs and outputs for Alice and Charlie. For the quantum correlation $P_Q^{14}$ of Eq. (30), for instance, one has $(V_{\mathrm{loc}},V_{\mathrm{biloc}})=(1,\frac{1}{2})$.

Figure 7

A trilocal scenario, with four parties returning outputs $x,a,b,y$ (here they do not receive any input). Under the assumption that $P(x,y)=P\left(x\right)P\left(y\right)>0$, the four-partite correlation $P(x,a,b,y)$ is trilocal if and only if the corresponding bipartite correlation $P(a,b|x,y)$ is local.

Figure 8

A three-locality scenario, where three parties receive states from three sources, forming a closed loop. Characterizing the (non)three locality of such a scenario remains an open problem.