Quantum Programs as Kleisli Maps

Furber and Jacobs have shown in their study of quantum computation that the category of commutative C∗-algebras and PU-maps (positive linear maps which preserve the unit) is isomorphic to the Kleisli category of a comonad on the category of commutative C∗-algebras with MIU-maps (linear maps which preserve multiplication, involution and unit). [3] In this paper, we prove a non-commutative variant of this result: the category ofC∗-algebras and PU-maps is isomorphic to the Kleisli category of a comonad on the subcategory of MIU-maps. A variation on this result has been used to construct a model of Selinger and Valiron’s quantum lambda calculus using von Neumann algebras. [1]