PhyslibAlpha.AlgebraicFramework.HilbertSpace.State.Density

Let $H$ be a complex Hilbert space and $\mathcal{L}(H)$ be the algebra of continuous linear operators on $H$. Given a positive operator $\rho \in \mathcal{L}(H)$ such that $\text{Tr}(\rho) = 1$ (commonly known as a density operator), this definition constructs the corresponding state $\omega$ on $\mathcal{L}(H)$. This state $\omega$ is a unital positive linear functional that maps an operator $x \in \mathcal{L}(H)$ to the complex number $\text{Tr}(x \rho)$.

Let $H$ be a complex Hilbert space and $\mathcal{L}(H)$ be the algebra of continuous linear operators on $H$. Let $\rho \in \mathcal{L}(H)$ be a positive operator with trace $\text{Tr}(\rho) = 1$. The unital positive linear map (state) $\omega$ associated with the density operator $\rho$ satisfies $$\omega(x) = \text{Tr}(x \rho)$$ for any operator $x \in \mathcal{L}(H)$.