Simple characterizations for commutativity of quantum weakest preconditions

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.

💡 Research Summary

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The paper addresses the problem of determining when two quantum weakest preconditions (wp) commute. In the quantum programming setting, a program P is modeled as a completely positive (CP) map (often given by Kraus operators {E_i}), and a quantum predicate M is a Hermitian operator satisfying 0 ≤ M ≤ I. The weakest precondition of M with respect to P is defined as wp(P,M) = ∑_i E_i† M E_i, which is the smallest predicate guaranteeing that after executing P the post‑condition M holds with the same probability.

Commutativity of wp(P,M) and wp(P,N) is important because it allows simultaneous reasoning about two predicates, analogous to the classical case where two assertions can be combined without ordering concerns. The existing literature (e.g., the 2007 IPL letter) gave sufficient conditions that are often cumbersome to verify in practice. This work proposes three new characterizations that are both necessary and sufficient under natural assumptions, and it demonstrates that a simple commutator condition