Polarization-transverse-spatial logical qubit entanglement purification using linear optics

We propose two types of logical qubit entanglement purification protocols (LEPPs) to reduce the effect of noises inside and outside the logical qubit subspace using linear optics, where logical qubit is encoded by single-photon polarization-transverse-spatial Bell state. For the noise outside of the logical qubit subspace, deterministic LEPP is presented to determine and correct the error-entangled state using error correction of the single-photon Bell state, resulting in the logical entangled state with unity fidelity. For the noise inside of the logical qubit subspace, probabilistic LEPP is presented to progressively increase the fidelity of the noisy logical entangled state using the parity and diagonal-basis measurements for the logical qubit. These two types of LEPPs have experimental feasibility with current technology, and the quantum operations of logical qubits in these protocols have fantastic applications in some other logical-encoding quantum information protocols.