  An assignment map is a mathematical operator that describes initial
system-environment states for open quantum systems. We reexamine the notion of
assignments, introduced by Pechukas, and show the conditions assignments can
account for correlations between the system and the environment, concluding
that assignment maps can be made linear at the expense of positivity or
consistency is more reasonable. We study the role of other conditions, such as
consistency and positivity of the map, and show the effects of relaxing these.
Finally, we establish a connection between the violation of positivity of
linear assignments and the no-broadcasting theorem.
