Realization and Characterization of a Two-Photon Four-Qubit Linear Cluster State

We report on the experimental realization of a four-qubit linear cluster state via two photons entangled both in polarization and linear momentum. This state was investigated by performing tomographic measurements and by evaluating an entanglement witness. By use of this state we carried out a novel nonlocality proof, the so-called “stronger two observer all-versus-nothing” test of quantum nonlocality.

Figure 1

Generation of the linear cluster state by a source of polarization-momentum hyperentangled two-photon state. The state $|\Xi ⟩=|{\Phi }^{-}⟩\otimes |{\psi }^{+}⟩$ corresponds to two separate two-qubit clusters. The HW acts as a CP thus generating the four-qubit linear cluster $|C_4⟩$.

Figure 2

Interferometer and measurement apparatus. (a) The mode pairs $r_A\mathrm{\text{−}}{\ell }_B$ and ${\ell }_A\mathrm{\text{−}}{r}_B$ are matched on the BS. The phase shifters ${\varphi }_A$ and ${\varphi }_B$ (thin glass plates) are used for the measurement of momentum observables. The polarization analyzers on each of BS output modes are shown. (b) Same configuration as in (a) with BS and glasses removed.

Figure 3

State characterization. (a) Coincidence rates versus path delay $\Delta x$ showing the interference pattern between the two pairs $r_A\mathrm{\text{−}}{\ell }_B$ and ${\ell }_A\mathrm{\text{−}}{r}_B$. The dip (peak) FWHM and the coherence time ($\sim 150\mathrm{fsec}$) of the photons are determined by the bandwidth (6 nm) of the interference filters used. (b) and (c) Tomographic reconstructions (real parts) of the polarization states corresponding to $|{\Phi }^{+}⟩$ and $|{\Phi }^{-}⟩$, respectively. The imaginary components are negligible. Typical uncertainties are 0.006 for the higher terms ($|HH⟩⟨HH|$, $|HH⟩⟨VV|$, $|VV⟩⟨HH|$, and $|VV⟩⟨VV|$) and 0.003 for the other terms.

Figure 4

Measurement setup for momentum (a),(b) and polarization (c) observables for photon $i$ ($i=A$, $B$). By the (a) setup we measure $x_i$ (${\varphi }_i=0$) and $y_i$ (${\varphi }_i=\frac{\pi }{2}$), while the (b) setup is used for measuring $z_i$. By the (c) setup we measure $X_i$ (${\theta }_Q=\frac{\pi }{4}$; ${\theta }_H=\frac{1}{8}\pi$, $\frac{3}{8}\pi$), $Y_i$ (${\theta }_Q=0$; ${\theta }_H=\frac{1}{8}\pi$, $\frac{3}{8}\pi$) and $Z_i$ (${\theta }_Q=0$; ${\theta }_H=0$, $\frac{\pi }{4}$), where ${\theta }_{H(Q)}$ is the angle between the HWP (QWP) optical axis and the vertical direction. The polarization analysis is performed contextually to $x_i$, $y_i$ (i.e., with BS and glass) or $z_i$ (without BS and glass), as shown by the dotted lines for BS and glass in (c).