PhyslibAlpha.AlgebraicFramework.HilbertSpace.State.Vector

Vector states

Vector states on spaces of continuous linear endomorphisms.

2 declarations

definition

Positive linear functional induced by $\psi$ (i.e., $x \mapsto \langle \psi, x \psi \rangle$ )

Given a vector $\psi$ in an inner product space $H$ over a field $\mathbb{k}$ , this definition constructs the positive linear functional on the algebra of continuous linear operators $\mathcal{B}(H)$ (denoted $H \to L[k] H$ ). The functional maps each operator $x$ to the scalar value $\langle \psi, x \psi \rangle_{\mathbb{k}}$ , which corresponds to the expectation value of $x$ in the state $\psi$ .

definition

Vector state $x \mapsto \langle \psi, x \psi \rangle$ associated with a unit vector $\psi$

#ofVec

For a unit vector $\psi$ in an inner product space $H$ over a field $\mathbb{k}$ (where $\|\psi\| = 1$ ), this definition constructs the unital positive linear map (state) on the algebra of continuous linear operators $\mathcal{B}(H)$ . The map sends each operator $x \in \mathcal{B}(H)$ to the scalar value $\langle \psi, x \psi \rangle$ .

PhyslibAlpha.AlgebraicFramework.HilbertSpace.State.Vector

Vector states

Positive linear functional induced by ψ\psiψ (i.e., x↦⟨ψ,xψ⟩x \mapsto \langle \psi, x \psi \ranglex↦⟨ψ,xψ⟩)

Vector state x↦⟨ψ,xψ⟩x \mapsto \langle \psi, x \psi \ranglex↦⟨ψ,xψ⟩ associated with a unit vector ψ\psiψ

Positive linear functional induced by $\psi$ (i.e., $x \mapsto \langle \psi, x \psi \rangle$ )

Vector state $x \mapsto \langle \psi, x \psi \rangle$ associated with a unit vector $\psi$