  Let Alice and Bob be able to make local quantum measurements and communicate
classically. The set of mathematically consistent joint probability assignments
(``states'') for such measurements is properly larger than the set of
quantum-mechanical mixed states for the Alice-Bob system. It is canonically
isomorphic to the set of positive (not necessarily completely positive) linear
maps Phi from the bounded linear operators on Alice's space to those on Bob's,
for which Tr Phi(I)=1. We review the fact that allowing classical communication
is equivalent to enforcing ``no-instantaneous-signalling'' (``no--influence'')
in the direction opposite the communication. We establish that in the subclass
of ``decomposable'' states, i.e. convex combinations of positive states with
"PTP" ones whose partial transpose is positive, the extremal states are just
the extremal positive and extremal PTP states. We show that two such states,
shared by the same pair of parties, cannot necessarily be combined as
independent states (their tensor product) if the full set of quantum operations
is allowed locally to each party. We use a framework of ``test spaces'' and
states on these, suited for exhibiting the analogies and deviations of
empirical probabilistic theories from classical probability theory. This leads
to a deeper understanding of analogies between quantum mechanics and Bayesian
probability theory. The existence of a ``most Bayesian'' quantum rule for
updating states after measurement, and its association with the situation when
information on one system is gained by measuring another, is a case of a
general proposition holding for test spaces combined subject to the
no-signalling requirement.
