Figure 1

(a) Graph associated to the hyperentangled state $|{\mathrm{HE}}_6\rangle$. Each set represents a photon and every vertex is associated to a qubit. Qubits $1$ and $4$ are encoded in the $E/I$ DOF, qubits $2$ and $5$ in polarization, and qubits $3$ and $6$ in the $r/\ell$ DOF. See text for further details. (b) Graph associated to the two-photon six-qubit linear cluster state $|{\mathrm{LC}}_6\rangle$. $|{\mathrm{LC}}_6\rangle$ can be obtained from $|{\mathrm{HE}}_6\rangle$ by applying two cz operations between qubits belonging to different DOFs.Reuse & Permissions

Figure 2

Setup of the experiment. (a) Source (green box) of the eight-mode hyperentangled state. A detailed description of the source is given in [22, 23, 29]. (b) Mode labeling. Upper (lower) modes correspond to Alice (Bob) photon. For each photon we indicate with $|r\rangle$ ($|\ell \rangle$) the right (left) modes and with $|I\rangle$ ($|E\rangle$) the internal (external) modes. We also show the two half-wave plates ($\lambda /2$) used to transform the hyperentangled state $|{\widetilde{\mathrm{HE}}}_6\rangle$ to the cluster state $|{\widetilde{\mathrm{LC}}}_6\rangle$. The $\lambda /2$ on the $I$ modes of photon $A$ is oriented at ${45}^{°}$ while the $\lambda /2$ on the $\ell$ modes of photon $B$ is oriented at $0^{°}$. (c) Measurement scheme. The momentum measurement setup consists of two chained interferometers, the first (BS${}_1$) measuring the $r/\ell$ qubit, while the second (BS${}_{2A}$ and BS${}_{2B}$) measuring the $I/E$ qubit. Polarization analysis is performed by standard wave plates and polarizing beam splitters (PBS). Path delays $\Delta x_1$ and $\Delta x_2$ are varied to obtain the optimal temporal superposition of the modes, respectively, in the first and second interferometers.Reuse & Permissions

Figure 3

Tomographic reconstruction of the four polarization states in Eq. (9) (real parts). The imaginary components are negligible. The corresponding theoretical Bell states are: (a) $|{\varphi }^{+}\rangle {}_{\pi }$, (b) $|{\varphi }^{-}\rangle {}_{\pi }$, (c) $|{\psi }^{+}\rangle {}_{\pi }$, (d) $|{\psi }^{-}\rangle {}_{\pi }$.Reuse & Permissions

Figure 4

Tomographic reconstruction of the four states encoded in the $r/\ell$ DOF (real parts). The imaginary components are negligible. The corresponding theoretical Bell states are: (1) and (3) $|{\psi }^{+}\rangle {}_{\mathbf{k}}$, (2) and (4) $|{\psi }^{-}\rangle {}_{\mathbf{k}}$.Reuse & Permissions

Figure 5

Tomographic reconstruction of the state encoded in the $E/I$ DOF (real part) corresponding to the $r_A-{\ell }_B$ spatial mode pair and the polarization state ${|}_{+}{|}_{+}$. The imaginary components is negligible. The corresponding theoretical state is $|{\varphi }^{+}\rangle {}_c$.Reuse & Permissions

Figure 6

Graph associated to the six-qubit horseshoe cluster state, equivalent to the generated linear cluster state.Reuse & Permissions

Figure 7

Measurement pattern I. A cross stands for a measurement in the basis $\{|0\rangle ,|1\rangle \}$, while $\alpha$ and $\beta$ indicate a measurement in the bases $B(\alpha )$ and $B(\beta )$. The output state is encoded in qubits 5 and 2. The circuit associated to the considered measurement is shown. We first indicate the circuit obtained by directly following the one-way rules and then the equivalent circuit composed by single-qubit gates and a two-qubit cnot gate. The gates labeled as “Change of basis” are due to the transformation between the computational and laboratory bases.Reuse & Permissions

Figure 8

Measurement patterns II, III, and IV and the corresponding circuit representations. Each circuit is composed by single-qubit gates and a two-qubit cnot gate.Reuse & Permissions

Figure 9

Graphic representation of the I-O matrices for the considered cnot operation. The sublabels indicate the pattern to which each matrix refers. The sets of input and output states are listed in Table  and can be read on the upper (input) and lower (output) axis.Reuse & Permissions

Figure 10

Tomographic reconstruction of the polarization entangled output state $|{\psi }_{\mathrm{out}}^{\prime }\rangle {}_{\mathrm{II}}$. Both the (a) real and (b) imaginary components are shown. The corresponding theoretical state is $-\frac{1}{\sqrt{2}}(|\mathit{HH}\rangle {}_{\mathit{AB}}-i|\mathit{VV}\rangle {}_{\mathit{AB}})$.Reuse & Permissions