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Hyperentanglement has attracted much attention for its fascinatingapplications in quantum communication. However, it is impossible to purify apair of photon systems in a mixed hyperentangled state with errors in twodegrees of freedom using linear optical elements only, far different from allthe existing entanglement purification protocols in a degree of freedom (DOF)for quantum systems. Here, we investigate the possibility of purifying aspatial-polarization mixed hyperentangled Bell state with the errors in boththe spatial mode and polarization DOFs, resorting to the nonlinear optics of anitrogen-vacancy (NV) center in a diamond embedded in a photonic crystal cavitycoupled to a waveguide. We present the first hyperentanglement purificationprotocol for purifying a pair of two-photon systems in a mixed hyperentangledBell state with the errors in two DOFs. Besides, we propose an efficienthyperentanglement concentration protocol for a partially hyperentangled Bellpure state, which has the maximal success probability in principle. These twoprotocols are useful in long-distance quantum communication withhyperentanglement

Hyperentanglement purification and concentration assisted by diamond NV centers inside photonic crystal cavities