Example: Constant laws without norm convergence

Choose distinct register factors ${\rho }_n$ and $p_n=|0⟩⟨0{|}_{{\rho }_n}$, extended by identity. The product-default state gives ${\omega }_0(p_n)=1$ for every $n$, but

Indeed joint basis vectors realize eigenvalues $1$ and $-1$ of the difference. The sequence is not norm Cauchy. The homes may converge to one physical point. Convergent outcome data therefore do not supply a norm-convergent observable.