Experimental Test of Compatibility-Loophole-Free Contextuality with Spatially Separated Entangled Qutrits

The physical impact and the testability of the Kochen-Specker (KS) theorem is debated because of the fact that perfect compatibility in a single quantum system cannot be achieved in practical experiments with finite precision. Here, we follow the proposal of A. Cabello and M. T. Cunha [Phys. Rev. Lett. 106, 190401 (2011)], and present a compatibility-loophole-free experimental violation of an inequality of noncontextual theories by two spatially separated entangled qutrits. A maximally entangled qutrit-qutrit state with a fidelity as high as $0.975\pm 0.001$ is prepared and distributed to separated spaces, and these two photons are then measured locally, providing the compatibility requirement. The results show that the inequality for noncontextual theory is violated by 31 standard deviations. Our experiments pave the way to close the debate about the testability of the KS theorem. In addition, the method to generate high-fidelity and high-dimension entangled states will provide significant advantages in high-dimension quantum encoding and quantum communication.

Figure 1

Scheme of the experiment. Two maximally entangled qutrits are distributed to spatially separated Alice and Bob. In each context, there is only one measurement performed by Alice ($D_1^A$ or $T_0^A$, $T_1^A$) and Bob ($D_0^B$) simultaneously on the qutrit that belongs to them. Spatial separation ensures the compatibility of the measurements on different qutrits.

Figure 2

Experimental setup. Part A is state preparation. A cw laser at 404 nm serves as the pump source. To generate qutrit-qutrit entanglement, the pump laser is separated into three paths by six half wave plates and two beam displacers, and then is directed into a 2 mm-thick $\beta$-barium borate (BBO) crystal. At BD1 the pump beam is split into two paths, which the vertically polarized ($V$) component is refracted in the $Z$ direction while the horizontally polarized ($H$) component is transmitted directly. After BD1 the pump state is $1/\sqrt{3}|H_0⟩+\sqrt{2}/\sqrt{3}|V_1⟩$. HWP3 in the down path rotates the polarization of the pump from $V$ polarized to the state $(|H⟩+|V⟩)/\sqrt{2}$. HWP2 at 0° is inserted into the up path to compensate the optical path difference between the up path and the down path. Then, the two paths pass through BD2 and are separated into three paths. HWP4 and HWP5 are used to rotate $H$ polarized to $V$ polarized, HWP6 is used as optical path difference compensator. After the three HWPs, the pump state is $(|V_0⟩+|V_1⟩+|V_2⟩)/\sqrt{3}$. The pump is then focused on three spots of the crystal to generate spatially entangled two photon state [31]. Thus the state $(|00⟩-|11⟩+|22⟩)/\sqrt{3}$ in the spatial mode is prepared if we encode the spatial mode of photons from the top to the bottom path as $|0⟩$, $|1⟩$, and $|2⟩$. Part B is the measurement $D_1^A$ and $D_0^B$ on Alice’s and Bob’s sides. Six fiber-coupled single-photon detectors $D_1-D_6$ are used to detect photons according to its coincidence counts $C_{i,j}$.

Figure 3

Graphical representation of the reconstructed density matrix of the two-photon state. Density matrix of the two qutrits is reconstructed from a set of 81 measurements represented by operators $u_i\otimes u_j$ (with $i,j=1,2,\dots ,9$) and $u_k=|{\mathrm{\Psi }}_k⟩⟨{\mathrm{\Psi }}_k|$. Kets $|{\mathrm{\Psi }}_k⟩$ for both idler photons and signal photons are selected from the following set: $|0⟩,|1⟩,|2⟩,(|0⟩+|1⟩)/\sqrt{2},(|1⟩+|2⟩)/\sqrt{2},(|1⟩+i|0⟩)/\sqrt{2},(|1⟩-i|2⟩)/\sqrt{2},(|0⟩+|2⟩)/\sqrt{2},(|0⟩+i|2⟩)/\sqrt{2}$. Detailed descriptions of tomographic measurements are presented in Refs. [34, 35].

Figure 4

Typical single-observable measurement devices. Experimental setups for measuring $D_1^A$, $T_0^A$, $T_1^A$ from top to bottom. Polarizing beam splitters, half wave plates, and beam displacers are used to construct the observables. Here we only explain how the setup for $T_0^A$ works. Consider that its eigenstate $(|1⟩+|2⟩)/\sqrt{2}$ is the input and that this photon state passes through the first BD directly. Then, the up path is rotated from $H$ polarized to $V$ polarized by a HWP set at 45°, while the down path is still $H$ polarized. After the second BD, this photon state is $(|H⟩+|V⟩)/\sqrt{2}$ and is rotated to $H$ polarized and passes through the PBS. Similarly, one can deduce the other two eigenstates as shown in the figure. $D_1^A$, $T_1^A$ can be analyzed similarly.