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

Abstract

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 proposal for how the evolution map ought to be defined. We summarize this standard proposal and then show that it sometimes fails to define a linear map or any map at all. Further, we show 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 postmortem of the standard proposal, 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; however, such inferences reflect not just the cause-effect relation between them---which is the basis for defining the correct evolution map---but also the common-cause relation. (ii) In a (retrospectively unnecessary) attempt to introduce variability in the input state, it inadvertently introduced variability in the inference map itself, and then tried to fit the input-output pairs associated to these different maps with a single map.