Commutativity of quantum weakest preconditions

The notion of quantum weakest precondition was introduced by D'Hondt and P. Panangaden [E. D'Hondt, P. Panangaden, Quantum weakest preconditions, Mathematical Structures in Computer Science 16 (2006) 429–451], and they presented a representation of weakest precondition of a quantum program in the operator-sum form. In this Letter, we give an intrinsic characterization of the weakest precondition of a quantum program given in a system-environment model. Furthermore, some sufficient conditions for commutativity of quantum weakest preconditions are presented.