  Probabilities enter quantum mechanics via Born's rule, the uniqueness of
which was proven by Gleason. Busch subsequently relaxed the assumptions of this
proof, expanding its domain of applicability in the process. Extending this
work to sequential measurement processes is the aim of this paper. Given only a
simple set of postulates, a probability measure is derived utilising the
concept of Liouville space and the most general permissible quantum channel
arises in the same manner. Super-Liouville space is constructed and a Bayesian
interpretation of this object is provided. An important application of the new
method is demonstrated, providing an axiomatic derivation of important results
of the BB84 protocol in quantum cryptography.
