  We develop a framework which unifies seemingly different extension (or
"joinability") problems for bipartite quantum states and channels. This
includes well known extension problems such as optimal quantum cloning and
quantum marginal problems as special instances. Central to our generalization
is a variant of the Choi-Jamiolkowski isomorphism between bipartite states and
dynamical maps which we term the "homocorrelation map": while the former
emphasizes the preservation of the positivity constraint, the latter is
designed to preserve statistical correlations, allowing direct contact with
entanglement. In particular, we define and analyze state-joining,
channel-joining, and local-positive joining problems in three-party settings
exhibiting collective UxUxU symmetry, obtaining exact analytical
characterizations in low dimension. Suggestively, we find that bipartite
quantum states are limited in the degree to which their measurement outcomes
may agree, while quantum channels are limited in the degree to which their
measurement outcomes may disagree. Loosely speaking, quantum mechanics enforces
an upper bound on the extent of positive correlation across a bipartite system
at a given time, as well as on the extent of negative correlation between the
state of a same system across two instants of time. We argue that these general
statistical bounds inform the quantum joinability limitations, and show that
they are in fact sufficient for the three-party UxUxU-invariant setting.
