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

Why initial system-environment correlations do not imply the failure of complete positivity: a causal perspective