Self-assisted complete maximally hyperentangled state analysis via the cross-Kerr nonlinearity

We present two complete maximally hyperentangled state analysis protocols for photons entangled in the polarization and spatial-mode degrees of freedom. The first protocol is a hyperentangled Bell state analysis scheme for two photons, and the second is a hyperentangled Greenberger-Horne-Zeilinger (GHZ) state analysis scheme for three photons. In each scheme, a set of mutually orthogonal hyperentangled basis states are completely and deterministically discriminated with the aid of cross-Kerr nonlinearities and linear optics. We also generalize the schemes to unambiguously analyze the $N$-photon hyperentangled GHZ state. Compared with previous protocols, our schemes greatly simplify the discrimination process and reduce the requirements on nonlinearities by using the measured spatial-mode state to assist in the analysis of the polarization state. These advantages make our schemes useful for practical applications in long-distance high-capacity quantum communication.

Figure 1

Schematic diagram of our complete HBSA protocol. The four cross-Kerr nonlinear interactions produce a phase shift of $\pm \theta$ on the coherent states $|{\alpha }_1\rangle$ and $|{\alpha }_2\rangle$ if photons are in corresponding spatial modes. The beam splitters (BSs) guide the photons from each input port to these two output ports with equal probabilities. The polarizing beam splitters (PBSs) at $0^{\circ }$ transmit horizontal polarized states while reflecting vertical polarized states. The PBSs at ${45}^{\circ }$ transmit $|+\rangle =\frac{1}{\sqrt{2}}\left(\right|H\rangle +|V\rangle )$, and they reflect $|-\rangle =\frac{1}{\sqrt{2}}\left(\right|H\rangle -|V\rangle )$. The dashed rectangle represents a single-photon Bell state measurement (SPBSM), which discriminates four single-photon Bell states completely. With the homodyne measurements on the two coherent states and two SPBSMs, 16 hyperentangled Bell states can be completely distinguished.

Figure 2

Schematic diagram of our complete HGSA protocol for three-photon hyperentangled GHZ states. Three coherent states are utilized to distinguish the spatial-mode GHZ states. The circles represent nonlinear interactions, and the phases inside the circles are the phase shift of the coherent states. SPBSM denotes a single-photon Bell state measurement as shown in Fig. 1, which can discriminate four single-photon Bell states with certainty.

Figure 3

Schematic diagram of our complete HGSA protocol for $N$-photon hyperentangled GHZ states. $N$ coherent states are used to discriminate the spatial-mode states without destroying the entanglement. Then $N$ SPBSMs are used to measure the polarization state with the help of the information about the spatial-mode state.