Entanglement purification of two-photon systems in multiple degrees of freedom

Hyperentanglement purification is a very important element for high-capacity long-distance quantum communication. We present an efficient hyperentanglement purification protocol (hyper-EPP) for nonlocal two-photon systems in three degrees of freedom (DoFs), including the polarization, the time-bin, and the spatial-mode DoFs. Our hyper-EPP is constructed with two steps resorting to polarization-time-spatial parity-check operation using the quantum nondemolition detector and SWAP gates, respectively. With this hyper-EPP, the bit-flip errors in the three DoFs can be corrected with high efficiency. The implementation of our hyper-EPP is contributed by a cross-Kerr nonlinearity between the signal photons and a coherent state via Kerr media, which could be achieved with current techniques. Moreover, some currently available optical elements are applied in the purification process, which offer facilities for the practical implementation. It is useful for long-distance high-capacity quantum communication with hyperentanglement in multiple DoFs.