Properties of a monad are identified so that its Kleisli category is a commutative effectus.

Effectuses have recently been introduced as categorical models for quantum computation, with probabilistic and Boolean (classical) computation as special cases. These â€˜probabilisticâ€™ models are called commutative effectuses. All known examples of such commutative effectuses are Kleisli categories of a monad. This paper answers the open question what properties a monad should satisfy so that its Kleisli category is a (commutative) effectus. The relevant properties are: strong affineness and partial additivity, together with some non-triviality conditions.