  The common wisdom in the field of quantum information theory is that when a
system is initially correlated with its environment, the map describing its
evolution may fail to be completely positive. If true, this would have
practical and foundational significance. We here demonstrate, however, that the
common wisdom is mistaken. We trace the error to the standard argument for how
the evolution map ought to be defined. We show that it sometimes fails to
define a linear map or any map at all and that these pathologies persist even
in completely classical examples. Drawing inspiration from the framework of
classical causal models, we argue that the correct definition of the evolution
map is obtained by considering a counterfactual scenario wherein the system is
reprepared independently of any systems in its causal past while the rest of
the circuit remains the same, yielding a map that is always completely
positive. In a post-mortem on the standard argument, we highlight two distinct
mistakes that retrospectively become evident (in its application to completely
classical examples): (i) the types of constraints to which it appealed are
constraints on what one can infer about the final state of a system based on
its initial state, where such inferences are based not just on the cause-effect
relation between them-which defines the correct evolution map-but also on the
common cause of the two; (ii) in a (retrospectively unnecessary) attempt to
introduce variability in the input state, it inadvertently introduced
variability in the inference map itself, then tried to fit the input-output
pairs associated to these different maps with a single map.
