Avoiding loopholes with hybrid Bell-Leggett-Garg inequalities

By combining the postulates of macrorealism with Bell locality, we derive a qualitatively different hybrid inequality that avoids two loopholes that commonly appear in Leggett-Garg and Bell inequalities. First, locally invasive measurements can be used, which avoids the “clumsiness” Leggett-Garg inequality loophole. Second, a single experimental ensemble with fixed analyzer settings is sampled, which avoids the “disjoint sampling” Bell inequality loophole. The derived hybrid inequality has the same form as the Clauser-Horne-Shimony-Holt Bell inequality; however, its quantum violation intriguingly requires weak measurements. A realistic explanation of an observed violation requires either the failure of Bell locality or a preparation conspiracy of finely tuned and nonlocally correlated noise. Modern superconducting and optical systems are poised to implement this test.

Figure 1

Schematic for the hybrid Bell-LGI using two pairs of sequential measurements for bounded properties. The measurements of $A_1$ and $A_2$ have noisy signals ${\alpha }_1$ and ${\alpha }_2$ that average to the range $[-1,1]$, while the remaining measurements of $B_1$ and $B_2$ have signals $b_1$ and $b_2$ constrained to the range $[-1,1]$. The correlator $C$ is averaged over realizations $\zeta$ of the joint preparation $P\left(\zeta \right)$. Postulating Bell locality and MR bounds this correlation to $|\langle C\rangle |\le 2$.

Figure 2

Bell-LGI correlator $\langle C\rangle$, where the $A_k$ eigenvalues $\pm 1$ are measured with signals ${\alpha }_k$ having Gaussian probability distributions (inset) with standard deviations ${\sigma }_k$. In the projective limit ${\sigma }_k\to 0$ the correlator converges to $\langle C\rangle \to 1/\sqrt{2}$, while in the weak limit ${\sigma }_k\to \infty$ it violates the local-macrorealistic (LMR) bound of 2 and converges to the quantum bound of $2\sqrt{2}$. The LMR bound is still violated for moderate $B_k$ visibility $v\in [0,1]$, while the efficiency $\eta \in [0,1]$ simply scales the ${\sigma }_k$ required to see the violations.

Figure 3

Bell-LGI correlator $\langle C\rangle$, where each $A_k$ is measured with an ancilla qubit, yielding the signal ${\alpha }_k=\pm 1/V_k$ with a discrete probability distribution (inset as a histogram for comparison with the Gaussian case inset in Fig. 2). The weak limit as the visibilities $V_k$ approach zero violates the LMR bound of 2 and converges to the quantum bound of $2\sqrt{2}$. The visibility $v\in [0,1]$ for $B_k$ decreases $\langle C\rangle$ similarly to Fig. 2, while the visibility $u\in [0,1]$ for the ancilla measurement scales the horizontal axis, $V_k\le u$.