PhyslibAlpha.ProbabilisticTheory.HilbertSpace.State.Vector

Given an inner product space $H$ over a field $\mathbb{k}$ (typically $\mathbb{R}$ or $\mathbb{C}$) and a unit vector $\psi \in H$ (satisfying $\|\psi\| = 1$), this definition constructs the associated vector state on the algebra of continuous linear operators $\mathcal{B}(H)$. This state is a unital positive linear functional that maps each operator $x \in \mathcal{B}(H)$ to the scalar value $\langle \psi, x \psi \rangle_{\mathbb{k}}$.

Let $H$ be an inner product space over $\mathbb{k}$ (where $\mathbb{k}$ is $\mathbb{R}$ or $\mathbb{C}$). For any unit vector $\psi \in H$ (such that $\|\psi\| = 1$) and any continuous linear operator $x \in \mathcal{B}(H)$, the value of the vector state $\omega_\psi$ applied to $x$ is given by the inner product $\langle \psi, x \psi \rangle$.