Failure of Local Realism Revealed by Extremely-Coarse-Grained Measurements

We show that failure of local realism can be revealed to observers for whom only extremely-coarse-grained measurements are available. In our instances, Bell’s inequality is violated even up to the maximum limit while both the local measurements and the initial local states under scrutiny approach the classical limit. Furthermore, we can observe failure of local realism when an inequality enforced by nonlocal realistic theories is satisfied. This suggests that locality alone may be violated while realism cannot be excluded for specific observables and states. Small-scale experimental demonstration of our examples may be possible in the foreseeable future.

Figure 1

(a) The numerically optimized Bell function $|B{|}_{\max }$ for the considered ETS as a function of the displacement $d$ and homodyne efficiency $\eta$ for $V=1000$. (b) The Bell function $|B{|}_{\max }$ as a function of $d$ and the variance of the initial thermal state $V$ when $\eta =0.05$. (c) Same as panel (b) but for the alternative form of ETS ${\rho }_{AB}^{\mathrm{alt}}$ defined in the manuscript. The horizontal plane in each figure indicates the classical limit 2.

Figure 2

The optimized Bell function $|B{|}_{\max }$ and the optimized Leggett function $|\mathcal{L}{|}_{\max }$ are presented, where $M=B$ for the solid curve and $M=\mathcal{L}$ for the dashed curve. (a) When $V=1$ and $\eta =1$ (pure entangled coherent states and perfect detectors), there is a range of $d$ where the Bell-CHSH inequality is violated while the Leggett’s inequality is satisfied. (b) When $V=1000$ (highly mixed ETSs), this range becomes larger. (c) The same effect can be obtained by decreasing $\eta$ to 0.03. (d) An example with $V=700$ and $\eta =0.05$ is given.