Quantum Key Recycling can share key more efficient than BB84

We calculate the key sharing rate of Lu et al.'s Quantum Key Recycling (QKR) protocol. The key sharing rate is another version of the key rate, but it can be calculated for both the Quantum Key Distribution (QKD) protocols and the QKR protocols. We define the key sharing rate in this study and compare the key sharing rate of the QKR protocol to the rate of the QKD protocols. We found Lu et al.'s QKR protocol can be used to share keys more efficiently than BB84 in some situations. We also compare the six-state version of Lu et al.'s QKR protocol to the six-state QKD protocol. The results of this study show the potential advantages of using pre-shared keys to replace the public discussion in quantum protocols.