PhyslibAlpha.ProbabilisticTheory.State.WeightEquivalence

Let $E$ be an order-unit space and $w$ be a weight defined on its positive cone. If $w$ is finite and normalized (i.e., $w(1) = 1$), this definition constructs the corresponding state $\varphi: E \to \mathbb{R}$. The state $\varphi$ is the unique unital positive linear map obtained by extending $w$ linearly from the positive cone to the entire space $E$.

Let $E$ be an order-unit space and $E^+$ be its positive cone. Given a state $s: E \to \mathbb{R}$ (represented as a unital positive linear map), this definition constructs the corresponding weight $w: E^+ \to [0, \infty]$. For any element $A \in E^+$, the weight is defined by $w(A) = s(A)$. Since $s$ is a positive linear map, $s(A)$ is guaranteed to be non-negative for all $A \in E^+$.

Let $E$ be an order-unit space and $E^+$ be its positive cone. Given a state $s: E \to \mathbb{R}$ (a unital positive linear map), let $w$ be the weight induced by $s$. For any element $A \in E^+$, the value of the weight $w(A)$ is given by $s(A)$ converted to an extended non-negative real number (an element of $[0, \infty]$).

Let $E$ be an order-unit space and $E^+$ be its positive cone. For any state $s$ on $E$ (defined as a unital positive linear map $s: E \to \mathbb{R}$) and any element $A \in E^+$, the weight induced by $s$ (denoted $s.\text{toWeight}$) satisfies the equality $(s.\text{toWeight}(A)).\text{toReal} = s(A)$. In other words, the real-valued evaluation of the weight induced by a state agrees with the value of the state itself on the positive cone.

Let $E$ be an order-unit space and $s: E \to \mathbb{R}$ be a state (a unital positive linear map). Let $w_s: E^+ \to [0, \infty]$ be the weight induced by $s$, defined by $w_s(A) = s(A)$ for all $A$ in the positive cone $E^+$. Then $w_s$ is finite and normalized.

Let $E$ be an order-unit space. There is an equivalence (bijection) between the set of finite normalized weights on the positive cone $E^+$ and the set of states $\mathcal{S}(\mathbb{R}, E)$ on $E$. A weight $w: E^+ \to [0, \infty]$ belongs to this subtype if it satisfies the property `IsState` (i.e., it is finite and normalized such that $w(1) = 1$), while a state $s \in \mathcal{S}(\mathbb{R}, E)$ is a unital positive linear map from $E$ to $\mathbb{R}$. This equivalence shows that restricting a state to the positive cone produces a finite normalized weight, and conversely, any such weight can be uniquely extended to a state.