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Coecke and Heunen described completely positive maps in dagger monoidalcategories and the {\sf CP}-infinity construction on these categories in orderto construct a category of arbitrary dimensional quantum processes. Thisarticle generalizes the ${\sf CP}$-infinity construction of dagger monoidalcategories to mixed unitary categories. Mixed unitary categories, on the onehand, generalize the (compact) dagger monoidal categories, and on the otherhand, accommodate arbitrary dimensional quantum processes, both withoutsacrificing the notion of dual objects. This means that the ${\sf CP}$-infinityconstruction for mixed unitary categories provides a suitable semantics forhigher-order quantum programming languages which employ arbitrary dimensionalstructures.  The existing results for the ${\sf CP}$-infinity construction are shown togeneralize to the new setting. In particular, the notion of environmentstructures generalizes to mixed unitary categories and it is shown that the${\sf CP}$-infinity construction for mixed unitary categories is characterizedby this generalized environment structure.

Complete Positivity for Mixed Unitary Categories