  Microwave photons have become very important qubits in quantum communication
as the first quantum satellite has been launched successfully. Therefore, it is
a necessary and meaningful task for ensuring the high security and efficiency
of microwave-based quantum communication in practice. Here, we present an
original polarization entanglement purification protocol for nonlocal microwave
photons based on the cross-Kerr effect in circuit quantum electrodynamics
(QED). Our protocol can solve the problem that the purity of maximally
entangled states used for constructing quantum channels will decrease due to
decoherence from environment noise. This task is accomplished by means of the
polarization parity-check quantum nondemolition (QND) detector, the
bit-flipping operation, and the linear microwave elements. The QND detector is
composed of several cross-Kerr effect systems which can be realized by coupling
two superconducting transmission line resonators to a superconducting molecule
with the N-type level structure. We give the applicable experimental parameters
of QND measurement system in circuit QED and analyze the fidelities. Our
protocol has good applications in long-distance quantum communication assisted
by microwave photons in the future, such as satellite quantum communication.
