PhyslibAlpha.AlgebraicFramework.Measurement.Basic

There is an equivalence between a measurement $m$ on an order-unit space $E$ with outcome labels in $\iota$ and the set of pairs $(I, f)$, where $I \subseteq \iota$ is a finite set of outcomes and $f: \iota \to \text{Effect } E$ is a function assigning an effect to each label, such that the sum of the effects over the finite set $I$ is equal to the identity (order unit) $1 \in E$, i.e., $\sum_{i \in I} f(i) = 1$.

Given a measurement $m$ on an order-unit space $E$ with a finite set of outcomes $I \subseteq \iota$ and a state $s$ (a unital positive linear functional $s: E \to \mathbb{R}$), the **outcome distribution** is the probability distribution $p$ on the set of outcomes $I$ defined by $p(i) = s(E_i)$ for each $i \in I$, where $E_i$ is the effect corresponding to outcome $i$. The resulting values are non-negative and sum to $1$, forming an element of the standard simplex $\Delta^I$.

Let $E$ be an order-unit space and $m$ be a measurement with a finite set of outcomes $I \subseteq \iota$ and associated effects $\{E_i\}_{i \in I}$. For any state $s$ (a unital positive linear functional $s: E \to \mathbb{R}$) and any outcome $i \in I$, the probability of outcome $i$ in the resulting outcome distribution is equal to the state $s$ evaluated on the corresponding effect $E_i$: $$(\text{outcomeDistribution } m, s)_i = s(E_i)$$

For a finite-outcome measurement $m$ on an order-unit space $E$ with a finite set of outcomes $I \subseteq \iota$, the **outcome map** is the function from the state space $\mathcal{S}(E)$ to the standard simplex $\Delta^I$ that assigns to each state $s$ the probability distribution defined by $s \mapsto (s(E_i))_{i \in I}$, where $E_i$ is the effect associated with the outcome $i \in I$.