Experimental Observation of Hardy-Like Quantum Contextuality

Contextuality is a fundamental property of quantum theory and a critical resource for quantum computation. Here, we experimentally observe the arguably cleanest form of contextuality in quantum theory [A. Cabello et al., Phys. Rev. Lett. 111, 180404 (2013)] by implementing a novel method for performing two sequential measurements on heralded photons. This method opens the door to a variety of fundamental experiments and applications.

Figure 1

Sequential measurements on the same photon. (a) The system is prepared in state $|\eta ⟩$ and then submitted to two nondemolition measurements, one after the other. The measurement $i$ is implemented by means of a unitary operation $U_i$ that maps the eigenstate of $i$ with eigenvalue 1 into the desired path, followed by an operation that stores the outcome in a detector, followed by the inverse unitary operation $U_i^{†}$ that rotates to the basis used in the state preparation. The measurement of $j$ is implemented similarly. (b) Our photonic implementation. The initial state and the measurements refer to the photon’s path degrees of freedom. The outcome of $i$ is encoded in the photon’s polarization. Then the unitary operation $U_j$ corresponding to $j$ is applied. Then the outcome of $i$ is decoded using polarizing beam splitters. The outcomes of $i$ and $j$ are given by the detector that clicks (since each detector corresponds to a combination of outcomes). There is no need to implement $U_j^{†}$, since no additional measurements will be performed.

Figure 2

Experimental setup for measuring observable 2 (in the first place) and observable 1, given by Eq. (5), on the initial state (4). After preparing state (4), the paths corresponding to $|0⟩$ and $|1⟩$ are injected into a Sagnac interferometer to control the phase difference. The phase shifter consists of two quarter wave plates and one HWP in between them. The result of the unitary operation $U_2$ is given by the Sagnac outputs together with the third path. Then the output of observable 2 is encoded by rotating the polarization in one of the paths with the help of a HWP at 45°. Then $U_2^{†}$ is applied to go back to the basis used in the state preparation, and the unitary operation $U_1$ is implemented in a similar way. The outcome of observable 2 is finally decoded using polarizing beam splitters (PBSs), so the single photon detector that clicks gives the outcomes of observables 2 and 1. All coincidence counts between the signal and idler photons are registered using an eight-channel coincidence logic with a time window of 1.7 ns. The number of detected photons was approximately $2\times 10^3$ per second and the total time used for each experimental configuration was 10 s.

Figure 3

Experimental setups for the unitary operations $U_i$ (in red) and $U_i^{†}$ (in blue) for $i=1,2,3$. They consist of combinations of BSs with $33:66$ and $50:50$ splitting ratios. The setups for $U_4$ and $U_4^{†}$ are the same as those for $U_3$ and $U_3^{†}$, respectively, but with the following relabeling of paths: $|0{⟩}_{U3}\to |1{⟩}_{U4}$, $|1{⟩}_{U3}\to |2{⟩}_{U4}$, and $|2{⟩}_{U3}\to |0{⟩}_{U4}$. The setups for $U_5$ and $U_5^{†}$ are the same as those for $U_2$ and $U_2^{†}$, respectively, but with the following relabeling of paths: $|0{⟩}_{U2}\to |1{⟩}_{U5}$, $|1{⟩}_{U2}\to |2{⟩}_{U5}$, and $|2{⟩}_{U2}\to |0{⟩}_{U5}$.