  Although it is impossible for a bit commitment protocol to be both
arbitrarily concealing and arbitrarily binding, it is possible for it to be
both partially concealing and partially binding. This means that Bob cannot,
prior to the beginning of the unveiling phase, find out everything about the
bit committed, and Alice cannot, through actions taken after the end of the
commitment phase, unveil whatever bit she desires. We determine upper bounds on
the degrees of concealment and bindingness that can be achieved simultaneously
in any bit commitment protocol, although it is unknown whether these can be
saturated. We do, however, determine the maxima of these quantities in a
restricted class of bit commitment protocols, namely those wherein all the
systems that play a role in the commitment phase are supplied by Alice. We show
that these maxima can be achieved using a protocol that requires Alice to
prepare a pair of systems in an entangled state, submit one of the pair to Bob
at the commitment phase, and the other at the unveiling phase. Finally, we
determine the form of the trade-off that exists between the degree of
concealment and the degree of bindingness given various assumptions about the
purity and dimensionality of the states used in the protocol.
