  Hyperentanglement is a promising resource in quantum information processing,
especially for increasing the channel capacity of long-distance quantum
communication. Hyperentanglement purification is an important method to obtain
high-fidelity nonlocal hyperentangled states from mixed hyperentangled states
in a long-distance quantum communication process with noisy channels. Here, we
present a two-step hyperentanglement purification protocol for nonlocal mixed
hyperentangled states with polarization bit-flip errors and spatial-mode
phase-flip errors, resorting to polarization-spatial phase-check quantum
nondemolition detectors and the quantum-state-joining method (QSJM). With QSJM,
the protocol can preserve the states that are discarded in the previous
hyperentanglement purification protocols. It has the advantage of a high
efficiency, and it is useful for improving the entanglement of photon systems
with several degrees of freedom in long-distance high-capacity quantum
communication.
