PhyslibAlpha.ProbabilisticTheory.Classical.EnsembleRefinement

Let $E$ be an order unit space. The property `EnsemblesRefine E` states that any two finite ensembles of weighted states with the same total admit a common refinement. Specifically, for any two finite families of positive linear functionals $\{\alpha_i\}_{i \in \mathcal{I}}$ and $\{\beta_j\}_{j \in \mathcal{J}}$ on $E$ such that $\sum_{i \in \mathcal{I}} \alpha_i = \sum_{j \in \mathcal{J}} \beta_j$, there exists a "table" of positive linear functionals $\{\gamma_{ij}\}_{i \in \mathcal{I}, j \in \mathcal{J}}$ satisfying: 1. The sum of each row matches the first ensemble: $\sum_{j \in \mathcal{J}} \gamma_{ij} = \alpha_i$ for all $i \in \mathcal{I}$. 2. The sum of each column matches the second ensemble: $\sum_{i \in \mathcal{I}} \gamma_{ij} = \beta_j$ for all $j \in \mathcal{J}$.

Let $E$ be a classical order unit space. Then ensembles of weighted states in $E$ admit a common refinement. That is, given two finite families of positive linear functionals $\{\alpha_i\}_{i \in \mathcal{I}}$ and $\{\beta_j\}_{j \in \mathcal{J}}$ on $E$ satisfying $$\sum_{i \in \mathcal{I}} \alpha_i = \sum_{j \in \mathcal{J}} \beta_j,$$ there exists a family of positive linear functionals $\{\gamma_{ij}\}_{i \in \mathcal{I}, j \in \mathcal{J}}$ such that $$\sum_{j \in \mathcal{J}} \gamma_{ij} = \alpha_i$$ for each $i \in \mathcal{I}$, and $$\sum_{i \in \mathcal{I}} \gamma_{ij} = \beta_j$$ for each $j \in \mathcal{J}$.

Let $E$ be an order unit space that satisfies the ensemble refinement property. Given two finite families of positive linear functionals $\{\alpha_i\}_{i \in \mathcal{I}}$ and $\{\beta_j\}_{j \in \mathcal{J}}$ on $E$ such that their totals are equal, i.e., $\sum_{i \in \mathcal{I}} \alpha_i = \sum_{j \in \mathcal{J}} \beta_j$, there exists a refinement table of positive linear functionals $\{\gamma_{ij}\}_{i \in \mathcal{I}, j \in \mathcal{J}}$ such that: 1. For each $i \in \mathcal{I}$, $\sum_{j \in \mathcal{J}} \gamma_{ij} = \alpha_i$. 2. For each $j \in \mathcal{J}$, $\sum_{i \in \mathcal{I}} \gamma_{ij} = \beta_j$.

Let $E$ be an order unit space that satisfies the ensemble refinement property (meaning any two finite ensembles of weighted states with the same total admit a common refinement). For any finite, non-empty family of elements $s = \{s_i\}_{i \in \iota}$ in $E$, and any two positive linear functionals $\alpha$ and $\beta$ on $E$, the upper envelope of $s$ is subadditive: $$\text{upperEnvelope}(s, \alpha + \beta) \leq \text{upperEnvelope}(s, \alpha) + \text{upperEnvelope}(s, \beta)$$

Let $E$ be an Archimedean order-unit space. If the state space $\mathcal{S}(E)$ is a simplex (meaning every state has a unique representation as a probability measure on the pure states), then $E$ is a classical space. In this context, being classical implies that the set of positive linear functionals on $E$ forms a lattice, such that any two positive functionals have a least upper bound (supremum) determined by the integration against the supremum of their corresponding measures.

If the state space of an Archimedean order-unit space $E$ is a simplex, then ensembles in $E$ refine. Specifically, for any two finite families of positive linear functionals $\{\alpha_i\}_{i \in \mathcal{I}}$ and $\{\beta_j\}_{j \in \mathcal{J}}$ on $E$ that prepare the same total state (i.e., $\sum_{i \in \mathcal{I}} \alpha_i = \sum_{j \in \mathcal{J}} \beta_j$), there exists a common refinement. This refinement is a collection of positive linear functionals $\{\gamma_{ij}\}_{i \in \mathcal{I}, j \in \mathcal{J}}$ such that the sum of each row matches the first ensemble ($\sum_{j \in \mathcal{J}} \gamma_{ij} = \alpha_i$) and the sum of each column matches the second ensemble ($\sum_{i \in \mathcal{I}} \gamma_{ij} = \beta_j$).