  Mayers, Lo and Chau argued that all quantum bit commitment protocols are
insecure, because there is no way to prevent an Einstein-Podolsky-Rosen (EPR)
cheating attack. However, Yuen presented some protocols which challenged the
previous impossibility argument. Up to now, it is still debated whether there
exist or not unconditionally secure protocols. In this paper the above
controversy is addressed. For such purpose, a complete classification of all
possible bit commitment protocols is given, including all possible cheating
attacks. Focusing on the simplest class of protocols (non-aborting and with
complete and perfect verification), it is shown how naturally a
game-theoretical situation arises. For these protocols, bounds for the cheating
probabilities are derived, involving the two quantum operations encoding the
bit values and their respective alternate Kraus decompositions. Such bounds are
different from those given in the impossibility proof. The whole classification
and analysis has been carried out using a "finite open system" approach. The
discrepancy with the impossibility proof is explained on the basis of the
implicit adoption of a "closed system approach"--equivalent to modeling the
commitment as performed by two fixed machines interacting unitarily in a
overall "closed system"--according to which it is possible to assume as "openly
known" both the initial state and the probability distributions for all secret
parameters, which can be then "purified". This approach is also motivated by
existence of unitary extensions for any open system. However, it is shown that
the closed system approach for the classification of commitment protocols
unavoidably leads to infinite dimensions, which then invalidate the continuity
argument at the basis of the impossibility proof.
