Experimental bilocality violation without shared reference frames

Nonclassical correlations arising in complex quantum networks are attracting growing interest, both from a fundamental perspective and for potential applications in information processing. In particular, in an entanglement swapping scenario a new kind of correlations arise, the so-called nonbilocal correlations that are incompatible with local realism augmented with the assumption that the sources of states used in the experiment are independent. In practice, however, bilocality tests impose strict constraints on the experimental setup and in particular to the presence of shared reference frames between the parties. Here, we experimentally address this point showing that false positive nonbilocal quantum correlations can be observed even though the sources of states are independent. To overcome this problem, we propose and demonstrate a scheme for the violation of bilocality that does not require shared reference frames and thus constitutes an important building block for future investigations of quantum correlations in complex networks.

Figure 1

Directed acyclic graphs [29] representing tripartite causal structures. Nodes represent random variables while arrows represent their causal relations. Blue circles show variables related to measurement outcomes, green squares represent measurement choice inputs, and orange blunt rectangles represent hidden variables. (a) Tripartite LHV model where the central node $B$ is made of two substations $B^A$ and $B^C$, driven by the same measurement choice $y$. (b) BLHV model with the two independent hidden variables ${\lambda }_1$ and ${\lambda }_2$ mediating the correlations. (c) Classical representation of the previous scenario when allowing for shared randomness between $A$ and $C$ (red rectangle). (d) Pictorial representation of the structure of the correlations in the three subpanels (a), (b), and (c).

Figure 2

Experimental setup. Two separated nonlinear crystals (EPR1 and EPR2) generate polarization-entangled photon pairs via a SPDC process. Photons 1 and 4 are directed to Alice's and Charlie's stations, respectively, where the analysis in polarization of a particular observable is performed by using a motorized HWP and a PBS. Photons 2 and 3 are directed to the two different substations $B^A$ and $B^C$ where an analogous analysis is performed.

Figure 3

Nonbilocal correlations from shared randomness. The green shaded area within the dotted lines shows the bilocal set in the $I,J$ plane, while the bold green line bounds the local set (that includes the bilocal one). The blue circle points show the experimental results for a fixed distribution (2), while the red square point represents the equal mixing of these two points (i.e., shared randomness). Error bars are due to Poissonian statistics.

Figure 4

Experimental test of bilocality with random reference frames between stations $B^C$ and $C$. (a) Distribution on the Bloch sphere of the performed unitaries for all the experimental points. Red cubes represent points chosen with a uniform pseudorandom distribution, while blue spheres represent random numbers changed for each event acquisitions. The orange pyramid depicts the identity unitary. (b) Amount of violation for all the different randomized unitaries. Given a unitary transformation $U_i$, green empty circles show the parameter $\mathcal{B}$ for those inequalities [out of 81, considering the different settings and symmetries of (1)] which violate the bilocality condition, except for the maximal violation which is represented with an orange triangle, red squares, or blue disks (corresponding, respectively, to the pyramid, cubes, and spheres of panel (a). Error bars indicate one standard deviation of uncertainty, due to Poissonian statistics. The orange straight (dashed) line shows the theoretical mean (minimum) violation expected for the maximum points (orange triangle, red squares, and blue disks) considering our subset of unitaries, while the red dot-dashed line represents their experimental mean value. The shaded areas display one standard deviation of uncertainty, both for the experimental mean value and for the theoretical expectations due to the experimental estimation of noise. The black dotted line indicates the bilocality violation bound.

Figure 5

Numerical analysis of the case when only $A$ and $B^A$ have a common reference frame. (a), (b) Mean (minimum) violation obtained for a growing population of $N$ random misaligned frames. Each point refers to a population which is totally independent from the other ones. The red line shows the violation bound (1.0), while the orange dashed line shows the minimum bound of $\sqrt[4]{2}$ derived in (7).

Figure 6

Numerical analysis of the case ${\mathcal{B}}^{2233}$ with our experimental subset of random unitaries. The green surface displays ${\mathcal{B}}_{\text{subset}}^{2233}$ for all the possible values of ${\gamma }_1$ and ${\gamma }_2$, while the orange plane shows the nonbilocality threshold.

Figure 7

Numerical analysis of the case when none of the parties have a common reference frame. (a),(b) Mean (minimum) violation obtained for a growing population of $N$ random misaligned frames. Each point refers to a population which is totally independent from the other ones. The red line shows the violation bound (1.0).

Figure 8

Numerical analysis of the case when no local calibration is ensured. (a) Probability density of obtaining a value $\mathcal{B}$. Different curves are related to an increasing number of settings (2,3,4) performed in stations $A$ and $C$. (b) Probability violation with respect to visibility of the Werner states ${\varrho }_{AB}$ and ${\varrho }_{BC}$ for the same cases.