1108.4607 (T. Lin)

Two simple characterizations of commutativity of quantum weakest
preconditions    [PDF]

T. Lin

In a recent letter by Ying et al. [Inf. Process. Lett. 104 (2007) 152-158], it has shown some sufficient conditions for commutativity of quantum weakest preconditions. However, the sufficient and necessary condition for quantum weakest preconditions remains open. This letter presents two simple characterizations of the commutativity of quantum weakest preconditions. We show that, (1). two quantum weakest preconditions $wp(\mathcal{E})(M)$ and $wp(\mathcal{E})(N)$ commute if and only if the product $wp(\mathcal{E})(M)\cdot wp(\mathcal{E})(N)$ (or, $wp(\mathcal{E})(N)\cdot wp(\mathcal{E})(M)$) is Hermitian; (2). two quantum weakest preconditions $wp(\mathcal{E})(M)$ and $wp(\mathcal{E})(N)$ commute if and only if there exists an Unitary matrix $U$ such that $U^{\dagger}wp(\mathcal{E})(M)U=\mathrm{diag}(\lambda_1,...,\lambda_n)$ and $U^{\dagger}wp(\mathcal{E})(N)U=\mathrm{diag}(\mu_1,...,\mu_n)$ where $\lambda_i$ and $\mu_i$ are the eigenvalues of $wp(\mathcal{E})(M)$ and $wp(\mathcal{E})(N)$, respectively.

View original: http://arxiv.org/abs/1108.4607