PhyslibAlpha.ProbabilisticTheory.Classical.FiniteDimensional

Let $E$ be an Archimedean order-unit space. If the state space $\mathcal{S}(E)$ is a geometric simplex, then the set of pure states (the extreme points of the state space) is finite.

Let $E$ be an Archimedean order-unit space. If the state space $\mathcal{S}(E)$ (the set of normalized positive continuous linear functionals on $E$) is a geometric simplex, then the pure states of $E$ are affinely independent as elements of the weak dual space $E^*$.

Let $E$ be an Archimedean order-unit space and let $P = \text{PureState } E$ be the set of its pure states, which is assumed to be finite. Let $\mu$ be a probability measure on $P$ and $\omega$ be a state in the state space $\mathcal{S}(E)$. If $\omega$ is the barycenter of $\mu$, such that for every $f \in E$ the evaluation $\omega(f)$ satisfies $$\omega(f) = \int_{P} k(f) \, d\mu(k),$$ then the state $\omega$, viewed as a point in the weak dual space $E^*$, is equal to the affine combination of the pure states $k \in P$ weighted by their masses $\mu(\{k\})$: $$\omega = \sum_{k \in P} \mu(\{k\}) k.$$

Let $E$ be an Archimedean order-unit space. If the state space $\mathcal{S}(E)$ is a geometric simplex—meaning it is the convex hull of a finite set of affinely independent pure states—then $\mathcal{S}(E)$ is a simplex state space. This implies that every state $\omega \in \mathcal{S}(E)$ can be uniquely decomposed as a mixture (convex combination) of pure states, where the weights correspond to unique barycentric coordinates.

Let $E$ be an Archimedean order-unit space. Let $s$ be a finite set of indices and $\{\phi_i\}_{i \in s}$ be a collection of states in the state space of $E$ (represented as elements of the weak dual space $E^*$). For any two sets of non-negative weights $w_{1,i}, w_{2,i} \in \mathbb{R}_{\ge 0}$, if the weighted sums of these states are equal in the weak dual space, i.e., $$\sum_{i \in s} w_{1,i} \phi_i = \sum_{i \in s} w_{2,i} \phi_i,$$ then the sums of the corresponding weighted states (viewed as positive linear maps) are also equal: $$\sum_{i \in s} (w_{1,i} \omega_i) = \sum_{i \in s} (w_{2,i} \omega_i),$$ where each $\omega_i$ is the unital positive linear map associated with the state $\phi_i$, and $c\omega$ denotes the weighted positive linear map defined by $(c\omega)(A) = c \cdot \omega(A)$ for all $A \in E$.

Let $E$ be an order unit space. If $E$ satisfies the ensemble refinement property (meaning that any two finite collections of positive linear functionals with the same sum admit a common refinement), then the set of pure states of $E$—defined as the extreme points of the state space in the weak dual space $E^*$—is affinely independent.

Let $E$ be a finite-dimensional Archimedean order-unit space. If the state space $\mathcal{S}(E)$ is a simplex (meaning every state $\omega \in \mathcal{S}(E)$ is uniquely represented as a mixture of pure states), then $\mathcal{S}(E)$ is a geometric simplex. A geometric simplex is the convex hull of finitely many affinely independent points, which in this case are the pure states of the system.

Let $E$ be a finite-dimensional Archimedean order-unit space. The state space $\mathcal{S}(E)$ is a simplex (meaning every state $\omega \in \mathcal{S}(E)$ is uniquely represented as a mixture of pure states) if and only if $\mathcal{S}(E)$ is a geometric simplex (meaning it is the convex hull of finitely many affinely independent pure states).

Let $E$ be an Archimedean order-unit space and $E^*$ its weak dual. Let $\mathcal{S}(E) \subseteq E^*$ be the state space of $E$. Suppose there exists a finite set $\iota$ and an affine isomorphism $e : E^* \cong_{\text{aff}} \mathbb{R}^\iota$ such that the image of the state space under $e$ is the standard simplex $\Delta^\iota$ (the set of vectors with non-negative coordinates that sum to $1$). Then the state space $\mathcal{S}(E)$ is a simplex, meaning every state in $\mathcal{S}(E)$ can be uniquely represented as a mixture of pure states.