Let $H$ be a [Hilbert space](/page/Hilbert%20Space), and let $\mathcal{L}(H)$ denote the $C^*$-algebra of bounded linear operators on $H$. A linear functional $\phi:\mathcal{L}(H)\to\mathbb C$ is a normal state if and only if there exists a positive trace-class operator $\rho\in\mathcal T(H)$ such that $\operatorname{Tr}(\rho)=1$ and

\begin{align*} \phi(a)=\operatorname{Tr}(\rho a) \end{align*}

for every $a\in\mathcal{L}(H)$.