Bell inequalities for position measurements

Bell inequalities for position measurements are derived using the bits of the binary expansion of position-measurement results. Violations of these inequalities are obtained from the output state of the nondegenerate optical parametric amplifier.

Figure 1

Calculation of $⟨s_{z,l}{s}_{z,l}^{\prime }⟩$ (solid line), $⟨s_{x,l}{s}_{x,l}^{\prime }⟩$ (dotted line), and $⟨s_{y,l}{s}_{\gamma ,l}^{\prime }⟩$ (dashed line) for $0.03⩽l⩽7.5$ and $r=0,0.5,1$, and 2 from top left to bottom right.

Figure 2

The expression (19) for lengths $0.03⩽l⩽7.5$ and $r=0,0.5,1$, and 2 from the lowest to the highest curve. The Bell inequality is violated whenever the value exceeds 2, e.g., when $l=1$ and $r=2$. The indicated values for $r=2$ will be used below.