PhyslibAlpha.ProbabilisticTheory.Classical.SeparableObservables

Let $E$ be an Archimedean order-unit space and let $(A_n)_{n \in \mathbb{N}}$ be a countable dense sequence of observables in $E$. For a given index $n \in \mathbb{N}$, this definition represents the continuous linear functional on the weak dual space $E^*$ of $E$ that maps an element $\omega \in E^*$ to the value $\omega(A_n)$. In the context of physical systems, this corresponds to the expectation value of the $n$-th observable in the dense sequence for a given state $\omega$.

Let $E$ be an Archimedean order-unit space and let $(A_n)_{n \in \mathbb{N}}$ be a countable dense sequence of observables in $E$. For any two states $x$ and $y$ in the state space of $E$, if the expectation values of all observables in the sequence are equal for both states—that is, $x(A_n) = y(A_n)$ for all $n \in \mathbb{N}$—then the states are identical ($x = y$).

Let $E$ be an Archimedean order-unit space. For every state $\omega \in \mathcal{S}(\mathbb{R}, E)$, there exists a probability measure $\mu$ on the state space $\text{stateSpace } E$ such that the barycenter of $\mu$ is $\omega$ and $\mu$ is concentrated on the pure states. That is, $\text{barycenter } \mu = \omega$ and for $\mu$-almost every $\phi$ in the state space, the state $\phi$ is pure.

Let $E$ be an Archimedean order-unit space. The set of pure states $\{\omega \in \text{stateSpace } E \mid \omega \text{ is pure}\}$ is a measurable set with respect to the Borel $\sigma$-algebra of the weak-star topology on the state space. A state is defined as pure if it is an extreme point of the convex state space.

Let $E$ be an Archimedean order-unit space. Let $\text{stateSpace } E$ be the set of normalized positive linear functionals on $E$ equipped with the Borel $\sigma$-algebra of the weak-star topology, and let $\text{PureState } E$ be the subset of pure states. The canonical inclusion map $\iota : \text{PureState } E \to \text{stateSpace } E$ is a measurable embedding.

Let $E$ be an Archimedean order-unit space, and let $\text{stateSpace } E$ denote its state space (the set of normalized positive continuous linear functionals). Let $P \subseteq \text{stateSpace } E$ be the set of pure states, and let $\iota: P \hookrightarrow \text{stateSpace } E$ be the inclusion map. If a measure $\mu$ on $\text{stateSpace } E$ is concentrated on the pure states—meaning that for $\mu$-almost every $\phi$, $\phi$ is a pure state—then the push-forward of the restriction (comap) of $\mu$ to $P$ along $\iota$ is equal to $\mu$: $$\iota_* (\iota^* \mu) = \mu$$

Let $E$ be an Archimedean order-unit space and let $\mathcal{S}(E)$ be its state space (the set of normalized positive continuous linear functionals in the weak dual $E^*$). Let $\mu$ be a probability measure on $\mathcal{S}(E)$. If $\mu$ is supported on the set of pure states (i.e., for $\mu$-almost every $\phi \in \mathcal{S}(E)$, the state $\phi$ is pure), then the pullback measure (comap) of $\mu$ under the inclusion map $\iota : \text{PureState } E \to \mathcal{S}(E)$ is a probability measure on the space of pure states.

Let $E$ be an Archimedean order-unit space with separable observables. For every state $\omega \in \mathcal{S}(E)$ (the set of unital positive linear maps from $E$ to $\mathbb{R}$), there exists a pure decomposition of $\omega$. Specifically, there exists a regular probability measure $\mu$ on the set of pure states of $E$ such that for every $f \in E$, the value of the state is given by the integral: $$\omega(f) = \int_{\text{PureState } E} \phi(f) \, d\mu(\phi)$$

For an Archimedean order-unit space $E$ representing a system with separable observables, the state space $\mathcal{S}(E)$ is a simplex if and only if the system satisfies the ensemble refinement property. This property, known as the **Choquet–Meyer theorem**, asserts that the uniqueness of representing a state as a mixture of pure states is equivalent to the ability to find a common refinement for any two finite ensembles of weighted states that have the same sum.