Figure 1

Preparation of the polarization-spatial path encoded states of single photons. The setup consists of a source of $H$-polarized single photons followed by a half wave plate (HWP) and a polarizing beam splitter (PBS), allowing any probability distribution of a photon in the paths $t$ and $r$. The wedge ($W$) placed in one of the paths adds an arbitrary phase shift between both paths. A HWP and a quarter wave plate (QWP) in each path allow us to rotate the outputs of the PBS to any polarization. Symbol definitions are given at the bottom of Fig. 2.Reuse & Permissions

Figure 2

Devices for measuring the nine observables (2). A measurement of $A$ requires only to distinguish between paths $t$ and $r$. For measuring $b$, note that its eigenstates are $(|t⟩\pm |r⟩)/\sqrt{2}$ and they need to be mapped to the paths $t$ and $r$; this is accomplished by interference with the help of an additional $50/50$ beam splitter (BS) and a wedge. The measurements of $a$ and $B$ are standard polarization measurements using a PBS and a HWP. Observables $C$, $c$, $\alpha$, $\beta$, and $\gamma$ are the product of a spatial path and a polarization observable ${\sigma }_i^s\otimes {\sigma }_j^p$. Each of these observables has a four-dimensional eigenspace, but since the observables need to be rowwise and columnwise compatible, only their common eigenstates can be used for distinguishing the eigenvalues. This implies that $C$, $c$, and $\gamma$ can be implemented as Bell measurements with different distributions of the Bell states. Similarly, $\alpha$ and $\beta$ are Bell measurements preceded by a polarization rotation. In this way $\gamma$ is compatible also with $\alpha$ and $\beta$.Reuse & Permissions

Figure 3

Setups for measuring the six sets of observables to test inequality (1). We explicitly describe the setup for measuring $C$, $A$, and $B$; the description of the other setups is obtained by replacing $C$, $A$, and $B$ with the corresponding observables. The seven boxes are single-observable measuring devices (see Fig. 2). The photon, prepared in a specific state, enters the device for measuring $C$ through the device’s input and follows one of the two possible outcomes. A detection of the photon in one of these outputs would make the measurement of the next observable impossible. Instead, we placed, after each of the two outputs of the $C$-measuring device, a device for measuring the second observable, $A$ (we thus used two identical $A$-measuring devices). Similarly, we also placed, after each of the four outputs of the $A$-measuring devices, a device for measuring the third observable, $B$ (we thus used four identical $B$-measuring devices). Note that we need to recreate the eigenstates of the measured observable before entering the next observable, since our single-observable measuring devices map eigenstates to a fixed spatial path and polarization. Finally, we placed a single-photon detector (D) after each of the eight outputs of the four $B$-measuring devices. An individual photon passing through the whole arrangement is detected only by one of the eight detectors, which indicates which one of the eight combinations of results for $C$, $A$, and $B$ is obtained.Reuse & Permissions

Figure 4

State independence of the violation of the inequality $\chi \le 4$. The value of $\chi$ was tested for 20 different quantum states: four pure states with maximum internal entanglement between the spatial path and polarization which would maximally violate a Clauser-Horne-Shimony-Holt–Bell-like inequality [27] (states $|{\psi }_1⟩-|{\psi }_4⟩$), one mixed state with partial internal entanglement which would violate a CHSH–Bell-like inequality (${\rho }_5$), one mixed state with partial internal entanglement which would not violate a CHSH–Bell-like inequality (${\rho }_6$), one mixed state without internal entanglement according to the Peres-Horodecki criterion [28, 29] (${\rho }_7$), 12 pure states without internal entanglement ($|{\psi }_8⟩-|{\psi }_{19}⟩$), and a maximally mixed state (${\rho }_{20}$). The explicit expression of each state is given in Table . The red solid line indicates the classical upper bound. The blue dashed line at 5.4550 indicates the average value of $\chi$ over all the 16 pure states.Reuse & Permissions

Figure 5

Correlation measurements of all terms in the inequality (1) for the states $|{\psi }_3⟩$ (a) and $|{\psi }_{14}⟩$ (b). The figures show experimentally estimated probabilities for detecting a photon in each of the eight detectors. A photon detection corresponds to certain values ($\pm 1$) for the three measured dichotomic observables. For example, the bar height at $+++$; $\alpha Aa$) represents the probability to obtain the results $\alpha$, $A$, $a=+1$, and similarly ($++-$; $\alpha Aa$) represents $\alpha$, $A=+1$ and $a=-1$. The expectation values for each measurement are also given.Reuse & Permissions