Complete hyperentangled-Bell-state analysis for quantum communication

It is impossible to unambiguously distinguish the four Bell states in polarization, resorting to linear optical elements only. Recently, the hyperentangled Bell state, the simultaneous entanglement in more than one degree of freedom, has been used to assist in the complete Bell-state analysis of the four Bell states. However, if the additional degree of freedom is qubitlike, one can only distinguish 7 from the group of 16 states. Here we present a way to distinguish the hyperentangled Bell states completely with the help of cross-Kerr nonlinearity. Also, we discuss its application in the quantum teleportation of a particle in an unknown state in two different degrees of freedom and in the entanglement swapping of hyperentangled states. These applications will increase the channel capacity of long-distance quantum communication.

Figure 1

Schematic diagram of the present HBSA protocol for spatial-mode entangled Bell states. This QND is used to distinguish the even-parity states $\{|{\varphi }^{\pm }\rangle {}_S\}$ from the odd-parity states $\{|{\psi }^{\pm }\rangle {}_S\}$. The action of first cross-Kerr nonlinearity puts a phase shift $\theta$ on the coherent probe beam if a photon appears in the mode coupled. The second cross-Kerr nonlinearity puts a phase shift with $-\theta$. After the nonlinear interactions, the probe beam picks up the phase shift $\theta$ or $-\theta$ if the state is $|a1b1\rangle$ or $|a2b2\rangle$, respectively. Otherwise, the states $|a1b2\rangle$ and $|a2b1\rangle$ pick up no phase shifts.

Figure 2

The second QND for the BSA in spatial modes. 50:50 BS acts as a Hadamard operation. ${\theta }_i$ ($i=1,2,3,4$) represent four different cross-Kerr nonlinear media with the phase shifts ${\theta }_i$. This QND is used to distinguish the four spatial modes $c1c2$, $d1d2$, $c1d2$, and $d1c2$ with the phase shifts ${\theta }_1+{\theta }_3$, ${\theta }_2+{\theta }_4$, ${\theta }_1+{\theta }_4$, and ${\theta }_2+{\theta }_3$, respectively. When the two photons appear at the spatial modes $c1c2$ or $d1d2$ ($c1d2$ or $d1c2$), they are originally in the Bell state $|{\varphi }^{+}\rangle {}_s$ ($|{\varphi }^{-}\rangle {}_s$) if the phase shift in the first QND is $\theta$ or $-\theta$. The result is kept for the state $|{\psi }^{+}\rangle {}_s$ ($|{\psi }^{-}\rangle {}_s$) when the phase shift in the first QND is zero. Completing the BSA with the first and the second QNDs, the outports of the two photons are determinate, which provides useful information for the analysis of the Bell states in the polarization degree of freedom.

Figure 3

Schematic diagram of the present HBSA protocol for Bell states in polarization. The state $|\mathit{HH}\rangle$ picks up the phase shift $\theta$ and $|\mathit{VV}\rangle$ picks up the phase shift $-\theta$. The states $|\mathit{HV}\rangle$ and $|\mathit{VH}\rangle$ pick up no phase shifts. That is, this setup is a parity-check device for the four Bell states in polarization.

Figure 4

The setup for distinguishing the relative phases of the Bell states in polarization. PBS, polarization beam splitter which is used to pass through $|H\rangle$ polarization photons and reflect $|V\rangle$ polarization photons. The wave plates $R_{45}$ rotate the horizontal and vertical polarizations by ${45}^{°}$, which accomplishes a Hadamard operation on polarization.

Figure 5

The procedure of teleportation of an unknown single-particle state in the polarization- and spatial-mode degrees of freedom with a hyperentanglement-state channel, resorting to HBSA.

Figure 6

Schematic diagram of hyperentanglement swapping in the polarization- and spatial-mode degrees of freedom. The initial hyperentangled states are prepared in nodes $\mathit{AB}$ and $\mathit{CD}$ (also the four photons). After Alice performs the HBSA on the two photons $\mathit{BC}$, Bob and Charlie can get the hyperentangled state between nodes $A$ and $D$. $a1b1$ ($a2b2$) are the different spatial modes for each hyperentangled state.