PhyslibAlpha.ProbabilisticTheory.Effect.Sharp

An effect $e$ is **sharp** if it is an extreme point of the effect interval $[0, 1]$. In other words, $e$ cannot be expressed as a nontrivial mixture (convex combination) $p a + (1 - p) b$ of two distinct effects $a$ and $b$ for any $p \in (0, 1)$.

The zero effect $0$ is sharp. In the context of the effect interval $[0, 1]$, this means that $0$ is an extreme point and cannot be expressed as a nontrivial mixture $p a + (1 - p) b$ of two distinct effects $a$ and $b$ for any $p \in (0, 1)$.

Let $e$ be an effect in the interval $[0, 1]$. If $e$ is sharp (meaning it is an extreme point of the effect interval that cannot be written as a nontrivial convex combination of distinct effects), then its complement $1 - e$ is also a sharp effect.

Let $E$ be a type equipped with a partial order, a zero element $0$, and a unit element $1$. For any effect $e \in [0, 1]$, $e$ is a sharp effect if and only if its complement $1 - e$ is a sharp effect. An effect is defined as sharp if it is an extreme point of the interval $[0, 1]$, meaning it cannot be written as a nontrivial convex combination $p a + (1 - p) b$ for $p \in (0, 1)$ and distinct effects $a, b$.

The certain outcome $1$, belonging to the set of effects $[0, 1]$, is a sharp effect. This means $1$ is an extreme point of the effect interval and cannot be expressed as a nontrivial convex combination of two distinct effects.