Experimental Violation of Two-Party Leggett-Garg Inequalities with Semiweak Measurements

We generalize the derivation of Leggett-Garg inequalities to systematically treat a larger class of experimental situations by allowing multiparticle correlations, invasive detection, and ambiguous detector results. Furthermore, we show how many such inequalities may be tested simultaneously with a single setup. As a proof of principle, we violate several such two-particle inequalities with data obtained from a polarization-entangled biphoton state and a semiweak polarization measurement based on Fresnel reflection. We also point out a nontrivial connection between specific two-party Leggett-Garg inequality violations and convex sums of strange weak values.

Figure 1

MR measurement schematic. An object pair is picked from an ensemble $\zeta$ at time $t_0$. At $t_1$ object 1 of the pair interacts with an imperfect detector for the property $A_1$, which reports a generalized value ${\alpha }_1$. At $t_2$ both objects interact with unambiguous detectors for the properties $B_1$ and $B_2$ that report values $b_1$ and $b_2$. The two-party LG correlation $C$ is constructed from the measured results.

Figure 2

Experimental setup. A 488 nm laser produces degenerate down-converted photon pairs. The polarization of the photon in the lower arm is rotated by 45° with a half wave plate, then undergoes semiweak polarization measurement in the $\{h,v\}$ basis using Fresnel reflection ($A_1$) that encodes the information in the resulting spatial modes, and is finally projected into the $\{\theta ,{\theta }_{\perp }\}$ basis with polarizers (pol.-$\theta$) set at angle $\theta$ ($B_1$). The attenuators (attn.) balance any remaining losses. The polarization of the photon in the upper arm is projected into the $\{h,v\}$ basis with another polarizer ($B_2$). The half and quarter wave plates are used for tomography of the input state; during data collection they are removed from the lower arm and used to switch between $h$ and $v$ polarization in the upper arm.

Figure 3

Real (left) and imaginary (right) parts of the reconstructed density matrix in the $\{h,v\}$ basis. Light gray (yellow) and dark gray (red) represent positive and negative values, respectively.

Figure 4

In all data plots, solid lines indicate theory and points indicate experimental data. (a) $⟨{\widehat{\sigma }}_z^{(1)}⟩$ (green, flat), ${}_{\theta }⟨{\widehat{\sigma }}_z^{(1)}⟩$ (red, decreasing), and ${}_{{\theta }_{\perp }}⟨{\widehat{\sigma }}_z^{(1)}⟩$ (blue, increasing). (b) ${}_{\theta ,h}⟨{\widehat{\sigma }}_z^{(1)}⟩$ (red, bottom right), and ${}_{{\theta }_{\perp },v}⟨{\widehat{\sigma }}_z^{(1)}⟩$ (blue, bottom left), violating negative bounds, unlike ${}_{{\theta }_{\perp },h}⟨{\widehat{\sigma }}_z^{(1)}⟩$ (orange, increasing), and ${}_{\theta ,v}⟨{\widehat{\sigma }}_z^{(1)}⟩$ (green, decreasing).

Figure 5

LGI correlation $⟨-{\sigma }_z^{(1)}-{\sigma }_z^{(1)}{\sigma }_{\theta }^{(1)}{\sigma }_z^{(2)}-{\sigma }_{\theta }^{(1)}{\sigma }_z^{(2)}⟩$ (red, squares) and the corresponding convex sum of the CAs ${}_{\theta ,h}⟨-{\widehat{\sigma }}_z^{(1)}⟩$ and ${}_{{\theta }_{\perp },v}⟨-{\widehat{\sigma }}_z^{(1)}⟩$ (blue, circles), both violating their upper bounds of 1 in the same domain of $\theta$. Compare to Fig. 4b and note that the LGI violation includes the region where the two CAs both exceed their bounds.

Figure 6

LGI correlations $⟨{\sigma }_z^{(1)}{\sigma }_z^{(2)}+{\sigma }_z^{(2)}{\sigma }_{\theta }^{(1)}-{\sigma }_z^{(1)}{\sigma }_{\theta }^{(1)}⟩$ (red, circles) and $⟨-{\sigma }_z^{(1)}{\sigma }_z^{(2)}+{\sigma }_z^{(2)}{\sigma }_{\theta }^{(1)}+{\sigma }_z^{(1)}{\sigma }_{\theta }^{(1)}⟩$ (blue, squares) violating their upper bounds of 1 for nearly the entire $\theta$ domain.