Valid conditions of the reference-frame-independent quantum key distribution

Reference-frame-independent quantum key distribution (RFI QKD) has been experimental demonstrated to be robust for slowly varying reference frames. However, the protocol fails when reference frame varies rapidly. Furthermore, the rigorous definition of “slowly varying” is still not clear. Here we analyze the relationship of the related parameters and show how to adjust these parameters to make RFI QKD to be valid. This work provides a clear valid condition for the RFI QKD protocol and opens a way for researchers and users to handle the RFI QKD process practically.

Figure 1

Transmission process of single photons. The change of reference frame makes $\theta$ vary during the transmission $\left(X_B^{{}^{\prime }}\ne X_B,Y_B^{{}^{\prime }}\ne Y_B\right)$. However, $Z$ is stable all the time.

Figure 2

The contrast between accumulative calculation and definite integral: (a) channel $X_A{X}_B$ and (b) channel $X_A{Y}_B$.

Figure 3

The changes of $C$ under definite integral.

Figure 4

The density function distribution of the photons detected in $X_A{X}_B$ and $X_A{Y}_B$ with $N_0=6\times {10}^4$ Hz.

Figure 5

Simulation data for secret key rate $r$ under the parameters $N={10}^4$ Hz, $\omega =\pi$, and $Q=4\%$. We choose different $T$ every 20 s. The results in the yellow region meet valid conditions, which show the robustness to the noise and produce relatively small errors. The gray regions are out of the valid range and show either large relative errors (the first 20 s) or small key rates (the last 20 s).