PhyslibAlpha.ProbabilisticTheory.State.StateSpace

Given an Archimedean order-unit space $E$ and a state $\omega$ (represented as a unital positive linear map in $\mathcal{S}(\mathbb{R}, E)$), this definition treats $\omega$ as a continuous linear functional in the strong dual space $E^*$. The continuity is established with respect to the order-unit norm $\|\cdot\|_1$, where the functional satisfies $|\omega(A)| \le \|A\|_1$ for all $A \in E$, implying its operator norm is at most $1$.

Let $E$ be an Archimedean order-unit space. For any state $\omega \in \mathcal{S}(\mathbb{R}, E)$ and any observable $A \in E$, the evaluation of $\omega$ as a continuous linear functional in the strong dual space $E^*$ (denoted by $\omega.\text{toStrongDual}$) at $A$ is equal to the value $\omega(A)$.

Let $E$ be an Archimedean order-unit space. The map $\text{toStrongDual} : \mathcal{S}(\mathbb{R}, E) \to E^*$ that associates each state (a unital positive linear map) $\omega \in \mathcal{S}(\mathbb{R}, E)$ with a continuous linear functional in the strong dual space $E^*$ is injective.

For any state $\omega \in \mathcal{S}(\mathbb{R}, E)$ on an Archimedean order-unit space $E$, the operator norm of $\omega$ when considered as a continuous linear functional in the strong dual space $E^*$ satisfies $\|\omega\| \le 1$. Here, the norm is taken with respect to the order-unit norm $\|\cdot\|_1$ on $E$.

Let $E$ be a nontrivial Archimedean order-unit space. For any state $\omega$ (a unital positive linear map in $\mathcal{S}(\mathbb{R}, E)$), the operator norm of $\omega$ viewed as a continuous linear functional in the strong dual space $E^*$ is equal to $1$, i.e., $\|\omega\| = 1$.

Let $E$ be an Archimedean order-unit space. For any two states $\omega, \varphi \in \mathcal{S}[\mathbb{R}, E]$ and any $t \in [0, 1]$, the continuous linear functional in the strong dual space $E^*$ corresponding to the convex mixture $t\omega + (1-t)\varphi$ is equal to the convex combination of their corresponding functionals, given by $t \cdot \omega^* + (1 - t) \cdot \varphi^*$, where $\omega^*$ and $\varphi^*$ denote the representations of $\omega$ and $\varphi$ as continuous linear functionals.

Given an Archimedean order-unit space $E$ with order unit $\mathbf{1}$, the state space is defined as the set of continuous linear functionals $f$ in the weak dual space $E^*$ that are normalized, such that $f(\mathbf{1}) = 1$, and positive, such that $f(A) \ge 0$ for all $A \in E$ with $A \ge 0$.

For an Archimedean order-unit space $E$, this function maps an element $\omega$ of the state space (defined as the set of normalized positive continuous linear functionals in the weak dual $E^*$) to its representation as a unital positive linear map $E \to \mathbb{R}$.

For an Archimedean order-unit space $E$, this function maps a state $\omega \in \mathcal{S}(\mathbb{R}, E)$ (modeled as a unital positive linear map) to its corresponding element in the state space $\text{stateSpace } E$, which is the set of normalized positive continuous linear functionals in the weak dual space $E^*$.

For an Archimedean order-unit space $E$, let $\omega$ be a state in the state space $\text{stateSpace } E$ (the set of normalized positive continuous linear functionals in the weak dual $E^*$). For any element $A \in E$, the evaluation of $\omega$ (viewed as a functional in the weak dual) at $A$ is equal to the evaluation of its representation as a unital positive linear map, $\text{toState}(\omega)$, at $A$. That is, $\omega(A) = \text{toState}(\omega)(A)$.

Let $E$ be an Archimedean order-unit space. For any state $\omega \in \mathcal{S}(\mathbb{R}, E)$, where $\mathcal{S}(\mathbb{R}, E)$ denotes the space of states modeled as unital positive linear maps from $E$ to $\mathbb{R}$, applying the conversion map $\text{ofState}$ followed by its inverse $\text{toState}$ returns the original state $\omega$. That is, $\text{toState}(\text{ofState}(\omega)) = \omega$.

Let $E$ be an Archimedean order-unit space. For any element $\omega$ in the state space $\text{stateSpace } E$ (the set of normalized positive continuous linear functionals in the weak dual $E^*$), converting $\omega$ to its representation as a state (a unital positive linear map) and then back to an element of the state space results in the original element $\omega$. That is, $\text{ofState}(\text{toState}(\omega)) = \omega$.

For an Archimedean order-unit space $E$, there is an equivalence between the state space $\text{stateSpace } E$ (the set of normalized positive continuous linear functionals in the weak dual $E^*$) and the space $\mathcal{S}(\mathbb{R}, E)$ of states (modeled as unital positive linear maps from $E$ to $\mathbb{R}$). This bijection is defined by the inverse mappings `toState` and `ofState`.

For an Archimedean order-unit space $E$, the mapping $\text{toState} : \text{stateSpace } E \to \mathcal{S}[\mathbb{R}, E]$, which identifies an element $\omega$ of the state space (defined as the set of normalized positive continuous linear functionals in the weak dual $E^*$) with its representation as a state (a unital positive linear map $E \to \mathbb{R}$), is injective.

For an Archimedean order-unit space $E$, the state space $\text{stateSpace } E$ is equipped with the measurable space structure defined by the Borel $\sigma$-algebra of the weak-star topology.

Let $E$ be an Archimedean order-unit space. The state space $\text{stateSpace } E$, considered as a subspace of the weak-star dual $E^*$, is a Borel space. This means that its predefined measurable space structure is exactly the Borel $\sigma$-algebra generated by the weak-star topology.

Let $E$ be an Archimedean order-unit space and let $\text{stateSpace } E$ be its state space equipped with the weak-star topology. For any element $A \in E$, the evaluation map $\omega \mapsto \omega(A)$ from the state space to $\mathbb{R}$ is continuous. In physical terms, this means that each expectation value depends continuously on the state.

In the state space of an Archimedean order-unit space $E$, a net of states $f_i$ converges to a state $\omega$ in the weak-star topology if and only if for every observable $A \in E$, the corresponding expectation values converge, i.e., $f_i(A) \to \omega(A)$.

For an Archimedean order-unit space $E$, the state space $\text{stateSpace } E$ is a closed subset of the weak dual space $E^*$ (the dual of $E$ equipped with the weak-star topology).

In an Archimedean order-unit space $E$ with order unit $\mathbf{1}$ and the associated order-unit norm $\|\cdot\|_1$, every functional $f$ in the state space $\text{stateSpace } E$ satisfies $\|f\| \le 1$, where $\|\cdot\|$ denotes the operator norm on the dual space $E^*$.

Let $E$ be an Archimedean order-unit space. The state space of $E$, denoted as $\text{stateSpace } E$, which consists of the normalized positive linear functionals in the weak dual space $E^*$, is compact with respect to the weak-star topology.

Let $E$ be an Archimedean order-unit space. The state space of $E$, denoted by $\text{stateSpace } E$ and consisting of all normalized positive linear functionals in the weak dual space $E^*$, is a compact topological space with respect to the weak-star topology.

Let $E$ be an Archimedean order-unit space. If $E$ is a separable topological space, then the state space $\text{stateSpace } E$ (the set of normalized positive linear functionals in the weak dual of $E$ equipped with the weak-* topology) is a metrizable space.

Let $E$ be an Archimedean order-unit space. The state space of $E$, denoted as $\text{stateSpace } E$, is the set of normalized positive linear functionals in the weak dual space $E^*$. This set is convex, meaning that any mixture (convex combination) of states is also a state.

Let $E$ be an Archimedean order-unit space. The range of the mapping `ofState`, which embeds states $\omega \in \mathcal{S}(\mathbb{R}, E)$ (unital positive linear maps) into the weak dual space $E^*$, is exactly the set $\text{stateSpace } E$ of normalized positive continuous linear functionals.

Let $E$ be an Archimedean order-unit space. A state $\omega$ in the state space $\text{stateSpace } E$ (the set of normalized positive continuous linear functionals in the weak dual $E^*$) is a pure state if and only if $\omega$ is an extreme point of the state space $\text{stateSpace } E$.