  Recently, two-photon six-qubit hyperentangled states were produced in
experiment and they can improve the channel capacity of quantum communication
largely. Here we present a scheme for the hyperentanglement purification of
nonlocal two-photon systems in three degrees of freedom (DOFs), including the
polarization, the first-longitudinal-momentum, and the second longitudinal
momentum DOFs. Our hyperentanglement purification protocol (hyper-EPP) is
constructed with two steps resorting to parity-check quantum nondemolition
measurement on the three DOFs and SWAP gates, respectively. With these two
steps, the bit-flip errors in the three DOFs can be corrected efficiently.
Also, we show that using SWAP gates is a universal method for hyper-EPP in the
polarization DOF and multiple longitudinal momentum DOFs. The implementation of
our hyper-EPP is assisted by nitrogen-vacancy centers in optical microcavities,
which could be achieved with current techniques. It is useful for long-distance
high-capacity quantum communication with two-photon six-qubit
hyperentanglement.

Quantum Physics