Polarization entanglement purification of nonlocal microwave photons based on the cross-Kerr effect in circuit QED

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.

Figure 1

(a) Schematic diagram of the cross-Kerr effect induced by coupling TLR $A$ (top, blue) and $B$ (bottom, red) to a superconducting molecule (middle, circle with $N$). The molecule can be controlled by external coils (left coils). The $N$-type level structure of the artificial molecule is shown in the right dashed line box. (b) The structure of superconducting quantum circuit for the molecule [56].

Figure 2

Schematic diagram of QND measurement of the total photon number of the two TLRs labeled with $A$ (i.e., $A_1$ and $A_2$). The two TLRs labeled with $B$ (i.e., $B_1$ and $B_2$) with the same decay rate ${\kappa }_2$ are the readout resonators and all the TLRs labeled with $A$ with the decay rate ${\kappa }_1$ are the storage resonators. The circle with $N$ stands for a superconducting molecule with the $N$-type level structure. The direction of the arrow represents the spread direction of the probe light. The elements labeled with a circular arrow in a big circle are circulators. $|\alpha \rangle$ represents the probe light. $|X\rangle \langle X|$ represents the homodyne measurement on the coherent state of the probe light.

Figure 3

Schematic diagram for the entanglement purification on two microwave-photon pairs. $S_1$ and $S_2$ are the two identical ideal entanglement sources for microwave-photon pairs. Two dashed boxes are two same-polarization parity-check QND detectors. PBS represents a polarizing beam splitter for microwave photons. The circles with a circular arrow stands for circulators. The QND measurement is given in Fig. 2. Two rectangular boxes labeled with $+/-$ signs are two measurements with the two diagonal bases $\left\{\right|\pm \rangle =\frac{1}{\sqrt{2}}\left(\right|H\rangle \pm |V\rangle )\}$.

Figure 4

Schematic diagram of our EPP for polarization-spatial entangled microwave-photon pairs. $S$ is the entanglement source for generating polarization-spatial entangled microwave-photon pairs. The two big dashed boxes are two polarization parity-check QND detectors. The two small dashed boxes are two couplers with the same PBS for microwave photons.

Figure 5

The fidelity of the state in the storage resonators with dissipation for different decay rates ${\kappa }_1^{-1}$. Here the decay rate of the readout resonators is ${\kappa }_2^{-1}\sim 10$ ns.

Figure 6

The fidelity of the state in the storage resonators with dissipation for different decay rates of the readout resonators ${\kappa }_2^{-1}$. Here the decay rate of the storage resonators is ${\kappa }_1^{-1}\sim 20\mu \mathrm{s}$.