Bilocal Bell Inequalities Violated by the Quantum Elegant Joint Measurement

Network Bell experiments give rise to a form of quantum nonlocality that conceptually goes beyond Bell’s theorem. We investigate here the simplest network, known as the bilocality scenario. We depart from the typical use of the Bell state measurement in the network central node and instead introduce a family of symmetric isoentangled measurement bases that generalize the so-called “elegant joint measurement.” This leads us to report noise-tolerant quantum correlations that elude bilocal variable models. Inspired by these quantum correlations, we introduce network Bell inequalities for the bilocality scenario and show that they admit noise-tolerant quantum violations. In contrast to many previous studies of network Bell inequalities, neither our inequalities nor their quantum violations are based on standard Bell inequalities and standard quantum nonlocality. Moreover, we pave the way for an experimental realization by presenting a simple two-qubit quantum circuit for the implementation of the elegant joint measurement and our generalization.

Figure 1

Bilocality scenario: Bob independently shares a “state” with Alice and Charlie, respectively. In a quantum experiment, these are independent, typically entangled quantum states ($|{\psi }^{-}⟩$), while in a bilocal model these are associated with independent local variables ($\alpha$ and $\gamma$).

Figure 2

The blue region represents the set of bilocal quantum correlations $p_{\mathrm{Q}}^{\theta =0}$ in the plane of visibilities $(V_1,V_2)$, with the dashed line in the figure inset showing the product of the visibilities on the boundary of this bilocal region. The red area is the part of the quantum region that can be detected as nonbilocal through the violation of our bilocal inequality [Eq. (9)].

Figure 3

Quantum circuit for implementing our family of generalized elegant joint measurements parameterized by $\theta$. A controlled-NOT gate is followed by a Hadamard rotation [$H=({\sigma }_1+{\sigma }_3)/\sqrt{2}$] on the control qubit, a controlled phase shift gate $R_{\pi /2-\theta }$, and a separate rotation of each qubit composed of $R_{\pi /2}$ and $H$. Finally, a measurement is performed in the basis $\{|00⟩,|01⟩,|10⟩,|11⟩\}$.