PhyslibAlpha.AlgebraicFramework.Measurement.FiniteOutcome

Given a finite family of effects $\{e_i\}_{i \in \iota}$ in an order unit space $E$ satisfying the completeness relation $\sum_{i \in \iota} e_i = 1$, this definition constructs a positive unital linear map $\Phi: (\iota \to \mathbb{R}) \to E$. For any function $f: \iota \to \mathbb{R}$, the map is defined by the linear combination $\Phi(f) = \sum_{i \in \iota} f(i) e_i$. This map is the unique linear extension of the assignment that maps each point mass (or indicator function) $\delta_i$ to its corresponding effect $e_i$.

Let $E$ be an order unit space and $\iota$ a finite index set. Given a family of effects $\{e_i\}_{i \in \iota}$ in $E$ satisfying the completeness relation $\sum_{i \in \iota} e_i = 1$, let $\Phi: (\iota \to \mathbb{R}) \to E$ be the channel (positive unital linear map) constructed from these effects via `toChannel`. For any function $f: \iota \to \mathbb{R}$, the value of the channel applied to $f$ is given by the linear combination: $$\Phi(f) = \sum_{i \in \iota} f(i) e_i$$

Let $E$ be an order unit space and $\iota$ a finite set of outcomes. Given a family of effects $\{e_i\}_{i \in \iota}$ in $E$ satisfying the completeness relation $\sum_{i \in \iota} e_i = 1$, let $\Phi: (\iota \to \mathbb{R}) \to E$ be the associated channel (a positive unital linear map) defined by $\Phi(f) = \sum_{j \in \iota} f(j) e_j$. For any $i \in \iota$, let $\delta_i$ denote the point mass (the indicator function) which takes the value $1$ at $i$ and $0$ elsewhere. Then the channel $\Phi$ maps the point mass $\delta_i$ to the corresponding effect $e_i$: $$\Phi(\delta_i) = e_i$$

Given a finite set of outcomes $\iota$ and a channel $M: (\iota \to \mathbb{R}) \to E$ (modeled as a positive unital linear map from the classical system $\mathbb{R}^\iota$ to an effect space $E$), the function `outcomeEffect` assigns to each outcome $i \in \iota$ the effect in $E$ given by $M(\delta_i)$, where $\delta_i$ is the indicator function (or point mass) that takes the value $1$ at $i$ and $0$ elsewhere. The result is a valid effect because $M$ is positive (ensuring $M(\delta_i) \geq 0$) and unital (ensuring $M(\delta_i) \leq M(\mathbb{1}) = 1$).

Let $\iota$ be a finite set of outcomes and $E$ be an effect space. For a channel $M : (\iota \to \mathbb{R}) \to E$ (represented as a positive unital linear map) and an outcome $i \in \iota$, the outcome effect of $M$ at $i$, when viewed as an element of $E$, is equal to the image of the indicator function $\delta_i$ under $M$, i.e., $M(\delta_i)$, where $\delta_i(j) = 1$ if $j = i$ and $0$ otherwise.

Let $\iota$ be a finite set of outcomes and $E$ be an effect space. For any channel $M : (\iota \to \mathbb{R}) \to E$, defined as a positive unital linear map from the classical system $\mathbb{R}^\iota$ to $E$, the sum of the outcome effects $M(\delta_i)$ over all $i \in \iota$ is equal to the unit $1$ of $E$, where $\delta_i$ is the indicator function (point mass) at outcome $i$.

For a finite set $\iota$ and an effect space $E$, this equivalence establishes a one-to-one correspondence between: 1. Channels $M: \mathbb{R}^\iota \to E$, which are positive unital linear maps from the classical $\iota$-outcome system to $E$. 2. Finite families of effects $\{e_i\}_{i \in \iota}$ in $E$ that satisfy the completeness relation $\sum_{i \in \iota} e_i = 1$. The correspondence maps a channel $M$ to the family of effects defined by $e_i = M(\delta_i)$, where $\delta_i$ is the point mass (indicator function) at outcome $i$. Conversely, a family $\{e_i\}$ is mapped to the channel $M(f) = \sum_{i \in \iota} f(i) e_i$.

Let $\iota$ be a finite set of outcomes and $E$ be an effect space. Given a channel $M : \mathbb{R}^\iota \to E$, defined as a positive unital linear map from the classical system $\mathbb{R}^\iota$ to $E$, the function `Measurement.ofChannel` constructs a finite-outcome measurement on $E$ with outcome labels $\iota$. The effects of this measurement are obtained by evaluating the channel on the standard basis of $\mathbb{R}^\iota$; specifically, the effect associated with outcome $i \in \iota$ is $M(\delta_i)$, where $\delta_i$ is the indicator function (point mass) at $i$. This collection of effects forms a valid measurement because the unitality of $M$ ensures that $\sum_{i \in \iota} M(\delta_i) = M(\sum_i \delta_i) = M(\mathbf{1}) = 1$.

Let $\iota$ be a finite set and $E$ be an effect space. Given a channel $M: \mathbb{R}^\iota \to E$ (a positive unital linear map) and a state $\omega$ on $E$, the probability of obtaining outcome $i \in \iota$ under the measurement induced by $M$ is equal to $\omega(M(\delta_i))$, where $\delta_i \in \mathbb{R}^\iota$ is the indicator function (point mass) at $i$.