Faithful entanglement purification for high-capacity quantum communication with two-photon four-qubit systems

As the hyperentanglement of photon systems presents lots of unique opportunities in high-capacity quantum networking, the hyperentanglement purification protocol (hyper-EPP) becomes a vital project work and the quality of its accomplishment attracts much attention recently. Here we present the first theoretical scheme of faithful hyper-EPP for nonlocal two-photon systems in two degrees of freedom (DOFs) by constructing several fidelity-robust quantum circuits for hyper-encoded photons. With this faithful hyper-EPP, the bit-flip errors in both the polarization and spatial-mode DOFs can be efficiently corrected and the maximal hyperentanglement in two DOFs could be in principle achieved by performing the hyper-EPP multiple rounds. Moreover, the fidelity-robust quantum circuits, parity-check quantum nondemolition detectors, and SWAP gates make this hyper-EPP works faithfully as the errors coming from practical scattering, in these quantum circuits, are converted into a detectable failure rather than infidelity. Furthermore, this hyper-EPP can be directly extended to purify photon systems entangled in single polarization or spatial-mode DOF and that hyperentangled in polarization and multiple-spatial-mode DOFs.