In this paper, we derive key-rate expressions for different quantum key distribution protocols. Our key-rate equations utilize multiple channel statistics, including those gathered from mismatched measurement bases - i.e., when Alice and Bob choose incompatible bases. In particular, we will consider an Extended B92 and a two-way semi-quantum protocol. For both these protocols, we demonstrate that their tolerance to noise is higher than previously thought - in fact, we will show the semi-quantum protocol can actually tolerate the same noise level as the fully quantum BB84 protocol. Along the way, we will also consider an optimal QKD protocol for various quantum channels. Finally, all the key-rate expressions which we derive in this paper are applicable to any arbitrary, not necessarily symmetric, quantum channel.
We consider a modified version of the BB84 quantum key distribution protocol in which the angle between two different bases is less than π 4. We show that the channel parameter estimate becomes the same as the original protocol with sufficiently transmitted qubits. On the other hand, the statistical correlation between bits transmitted in one basis and those received in the other basis becomes stronger as the angle between two bases becomes narrower. If the angle is very small, the statistical correlation between bits transmitted in one basis and those received in the other basis is as strong as those received in the same basis as the transmitting basis, which means that the modified protocol can generate almost twice as long secret key as in the original protocol, provided that Alice and Bob choose two different bases with almost the same probability. We also point out that the reverse reconciliation often gives a different amount of secret key to the direct reconciliation over Pauli channels with our modified protocol.