Simple characterizations for commutativity of quantum weakest preconditions

TL;DR

This paper offers new, simplified characterizations for when quantum weakest preconditions commute, advancing understanding of their algebraic properties and highlighting the complexity of their commutativity conditions.

Contribution

It introduces alternative simple criteria for quantum weakest precondition commutativity and discusses the difficulty of characterizing this property via operator commutators.

Findings

Provided new theorems and propositions on quantum weakest precondition commutativity.

Showed the complexity of characterizing commutativity through operator commutators.

Compared new characterizations with existing sufficient conditions.

Abstract

In a recent letter [Information Processing Letters~104 (2007) 152-158], it has shown some sufficient conditions for commutativity of quantum weakest preconditions. This paper provides some alternative and simple characterizations for the commutativity of quantum weakest preconditions, i.e., Theorem 3.1, Theorem 3.2 and Proposition 3.3 in what follows. We also show that to characterize the commutativity of quantum weakest preconditions in terms of $[M,N]$ ($=MN-NM$) is hard in the sense of Proposition 4.1 and Proposition 4.2.