Complete Positivity for Mixed Unitary Categories

Coecke and Heunen described completely positive maps in dagger monoidal categories and the {\sf CP}-infinity construction on these categories in order to construct a category of arbitrary dimensional quantum processes. This article generalizes the ${\sf CP}$-infinity construction of dagger monoidal categories to mixed unitary categories. Mixed unitary categories, on the one hand, generalize the (compact) dagger monoidal categories, and on the other hand, accommodate arbitrary dimensional quantum processes, both without sacrificing the notion of dual objects. This means that the ${\sf CP}$-infinity construction for mixed unitary categories provides a suitable semantics for higher-order quantum programming languages which employ arbitrary dimensional structures. The existing results for the ${\sf CP}$-infinity construction are shown to generalize to the new setting. In particular, the notion of environment structures generalizes to mixed unitary categories and it is shown that the ${\sf CP}$-infinity construction for mixed unitary categories is characterized by this generalized environment structure.