  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.
