Experimental implementation of an eight-dimensional Kochen-Specker set and observation of its connection with the Greenberger-Horne-Zeilinger theorem

For eight-dimensional quantum systems there is a Kochen-Specker (KS) set of 40 quantum yes-no tests that is related to the Greenberger-Horne-Zeilinger (GHZ) proof of Bell's theorem. Here we experimentally implement this KS set using an eight-dimensional Hilbert space spanned by the transverse momentum of single photons. We show that the experimental results of these tests violate a state-independent noncontextuality inequality. In addition, we show that, if the system is prepared in states that are formally equivalent to a three-qubit GHZ and W states, then the results of a subset of 16 tests violate a noncontextuality inequality that is formally equivalent to the three-party Mermin's Bell inequality, but for single eight-dimensional quantum systems. These experimental results highlight the connection between quantum contextuality and nonlocality for eight-dimensional quantum systems.

Figure 1

(a) The Kernaghan and Peres KS set of 40 quantum yes-no tests. Each test is represented by a straight line. The results of two tests cannot both be 1 when the lines are parallel or when there is a dot in their intersection. The initial state and the 16 KS tests violating the Bell inequality are indicated by a green line (the upper horizontal line) and 16 red (gray) lines, respectively. (b) Mermin's proof of state-independent quantum contextuality. There are ten observables represented as nodes in a pentagram. $ZXX$ denotes observable ${\sigma }_z\otimes {\sigma }_x\otimes {\sigma }_x$ on a system of three qubits, where ${\sigma }_z$ represents the Pauli $z$ matrix. Compatible observables are in the same line. Each of the 40 KS tests in (a) is the projection on a common eigenstate of 4 of the observables in (b). For example, states 1–8 in (a) are the common eigenstates of the observables in the horizontal line in (b) and similarly for the other lines. The impossibility of assigning noncontextual results $-1$ or 1 to the observables in (b) in agreement with the predictions of QT leads to the impossibility of assigning noncontextual results 1 or 0 to the yes-no tests in (a).

Figure 2

Experimental setup. A CW laser, an AOM, and calibrated attenuators (not shown for clarity) are used to produce faint optical pulses. The weak coherent states are transversally expanded by a telescope and sent through four transmissive SLMs placed in series, and with each LCD at the image plane of the previous one. SLM1 and SLM2 are used to prepare initial eight-dimensional quantum states encoded in the linear transverse momentum of single photons. The generated states are then used to test inequalities (1) and (4) after performing, on each of them, the 40 KS vector projections. The KS projections are carried out using SLM3, SLM4, and a pointlike APD (see the main text for details).

Figure 3

Probabilities of result 1 for the 40 KS tests in $\Sigma$ when the initial state is the GHZ state. The 40 KS tests are grouped in five bases, as in Fig. 1. The theoretical predictions for an ideal experiment are shown in the upper right corner. $F$ is the similarity of the recorded and theoretical probability distributions.

Figure 4

(a) Experimental results for $\Sigma$. The dotted line “Noncontextual limit (ideal)” indicates the maximum possible value of $\Sigma$ for noncontextual hidden variable theories in an ideal experiment in which the exclusivity relations between the KS tests are perfect. The dotted line “Noncontextual limit” indicates the value when experimental imperfections are taken into account, and similarly for the quantum limits. (b) Experimental results for $S$. The measured violations for the GHZ and W states are within the theoretical predictions for an ideal experiment, namely, $S_{\mathrm{GHZ}}=4.0$ and $S_W=3.5$, respectively.

Figure 5

(a, c) The measured value of $\Sigma$ for a certain number of detected pulses for the $|\mathrm{GHZ}\rangle$ and $|W\rangle$ state, respectively. (b, d) The measured value of the NC Mermin inequality violation for a certain number of detected pulses for the $|\mathrm{GHZ}\rangle$ and $|W\rangle$ state, respectively.

Figure 6

(a–c) The measured value of $\Sigma$ for a certain number of detected pulses for the $|{\Psi }_{\mathrm{prod}}\rangle$, $|\eta \rangle$, and $|\beta \rangle$ states, respectively.

Figure 7

(a–d) The probabilities of result 1 for the 40 KS tests performed for the $|W\rangle$, $|{\Psi }_{\mathrm{prod}}\rangle$, $|\eta \rangle$, and $|\beta \rangle$ states, respectively. The fidelities, F, between the recorded and the expected probability distributions are $0.97\pm 0.01$, $0.92\pm 0.02$, $0.98\pm 0.01$, and $0.95\pm 0.02$, respectively.

Figure 8

(a–d) The projection probabilities for the KS vectors orthogonal to the $|{\Psi }_{{\mathrm{v}}_{27}}\rangle$, $|{\Psi }_{{\mathrm{v}}_{34}}\rangle$, $|{\Psi }_{{\mathrm{v}}_{36}}\rangle$, and $|{\Psi }_{{\mathrm{v}}_{40}}\rangle$ states, respectively. The mean values of the probabilities shown are $0.016\pm 0.002,0.011\pm 0.001,0.011\pm 0.001$, and $0.017\pm 0.001$, respectively.