  It is well known that no quantum bit commitment protocol is unconditionally
secure. Nonetheless, there can be non-trivial upper bounds on both Bob's
probability of correctly estimating Alice's commitment and Alice's probability
of successfully unveiling whatever bit she desires. In this paper, we seek to
determine these bounds for generalizations of the BB84 bit commitment protocol.
In such protocols, an honest Alice commits to a bit by randomly choosing a
state from a specified set and submitting this to Bob, and later unveils the
bit to Bob by announcing the chosen state, at which point Bob measures the
projector onto the state. Bob's optimal cheating strategy can be easily deduced
from well known results in the theory of quantum state estimation. We show how
to understand Alice's most general cheating strategy, (which involves her
submitting to Bob one half of an entangled state) in terms of a theorem of
Hughston, Jozsa and Wootters. We also show how the problem of optimizing
Alice's cheating strategy for a fixed submitted state can be mapped onto a
problem of state estimation. Finally, using the Bloch ball representation of
qubit states, we identify the optimal coherent attack for a class of protocols
that can be implemented with just a single qubit. These results provide a tight
upper bound on Alice's probability of successfully unveiling whatever bit she
desires in the protocol proposed by Aharonov et al., and lead us to identify a
qubit protocol with even greater security.
