PhyslibAlpha.ProbabilisticTheory.Measurement.Binary

For an effect $e$ in an ordered additive commutative group $E$ with unit $1$, and a boolean value $b \in \{\text{true}, \text{false}\}$, this function returns $e$ if $b = \text{true}$ and the complement $1 - e$ if $b = \text{false}$.

Let $E$ be an ordered additive commutative group with a unit element $1$. For any effect $e \in E$ (where $0 \le e \le 1$), the sum of its binary atoms over all boolean outcomes $b \in \{\text{true}, \text{false}\}$ is equal to the unit element $1$. That is, $\sum_{b \in \{\text{true}, \text{false}\}} \text{binaryAtom}(e, b) = 1$.

Given an effect $e$ in an ordered additive commutative group $F$ with unit $1$ (meaning $e \in F$ such that $0 \le e \le 1$), the binary measurement associated with $e$ is a measurement on the boolean outcome space $\{\text{true}, \text{false}\}$. This measurement is defined such that the effect corresponding to the outcome $\text{true}$ is $e$, and the effect corresponding to the outcome $\text{false}$ is the complement $1 - e$.

Let $F$ be an ordered additive commutative group with unit $1$. For any effect $e \in F$ (where $0 \le e \le 1$), let $\text{binaryMeasurement}(e)$ be the measurement on the outcome space $\{\text{true}, \text{false}\}$ associated with $e$. Then, the effect corresponding to the measurable set $\{\text{true}\}$ under this measurement is equal to $e$. That is, $\text{binaryMeasurement}(e)(\{\text{true}\}) = e$.

For any effect $e$ (an element satisfying $0 \le e \le 1$) in an ordered additive commutative group $F$ with unit $1$, the binary measurement associated with $e$ assigns the complement effect $1 - e$ to the measurable set containing the outcome $\{\text{false}\}$.