PhyslibAlpha.AlgebraicFramework.OrderUnit.State.Basic

For a field $\mathbb{k}$ and an ordered $\mathbb{k}$-vector space $A$, the notation $\mathcal{P}[\mathbb{k}, A]$ denotes the set of positive linear functionals from $A$ to its base field $\mathbb{k}$.

For an ordered complex vector space $A$, let $\mathcal{P}[A]$ denote the set of all positive linear functionals from $A$ to the complex numbers $\mathbb{C}$. A linear functional $f: A \to \mathbb{C}$ is positive if it maps the positive cone of $A$ to the non-negative real numbers $[0, \infty) \subset \mathbb{C}$.

The notation $\mathcal{S}[\mathbb{k}, A]$ denotes the state space of an ordered $\mathbb{k}$-vector space $A$ that possesses a unit element. A state in this space is a normalized positive linear functional $f: A \to \mathbb{k}$. Specifically, $f$ is a state if it is linear, maps positive elements of $A$ to non-negative values in $\mathbb{k}$, and maps the unit element of $A$ to $1$.

For an ordered complex vector space $A$ with a unit, the notation $\mathcal{S}[A]$ denotes its state space, which is the set of all normalized positive linear functionals $\omega: A \to \mathbb{C}$. These are linear maps that send positive elements of $A$ to non-negative real numbers and the unit element of $A$ to $1$.

For an ordered real vector space $E$ with a unit element $1$, let $\mathcal{S}[\mathbb{R}, E]$ be the state space consisting of normalized positive linear functionals $\omega: E \to \mathbb{R}$ such that $\omega(1) = 1$. Given a state $\omega$ and an effect $e \in E$ (where $0 \leq e \leq 1$), this function returns the value $\omega(e)$ as an element of the unit interval $[0, 1]$. In physical terms, this represents the probability that the effect $e$ occurs when the system is in state $\omega$.

Let $E$ be an ordered real vector space with a unit element $1$. For any state $\omega \in \mathcal{S}[\mathbb{R}, E]$ and any effect $e \in E$ (an element such that $0 \le e \le 1$), the value of $\omega$ evaluated on $e$ as an element of the unit interval $[0, 1]$ is equal to the real number $\omega(e)$.

Let $E$ be an ordered real vector space with a unit element $1$. For any state $\omega \in \mathcal{S}[\mathbb{R}, E]$ (a normalized positive linear functional such that $\omega(1) = 1$) and any effect $e \in E$ (an element satisfying $0 \le e \le 1$), the probability assigned to the complement effect $1 - e$ is the complement of the probability assigned to $e$ within the unit interval $[0, 1]$. Mathematically, this is expressed as: $$\omega(1 - e) = 1 - \omega(e)$$

Let $E$ be an ordered real vector space with a unit element $1$. Let $\mathcal{S}[\mathbb{R}, E]$ be the state space consisting of normalized positive linear functionals $\omega: E \to \mathbb{R}$ such that $\omega(1) = 1$. For any state $\omega \in \mathcal{S}[\mathbb{R}, E]$ and any two orthogonal effects $e, f \in E$ (elements where $0 \le e, f \le 1$ and their sum is also an effect), the probability of the sum of the effects is the sum of the individual probabilities: $$\omega(e + f) = \omega(e) + \omega(f)$$

Let $E$ be an ordered real vector space with unit element $1$. Let $\mathcal{S}[\mathbb{R}, E]$ denote the state space of $E$, consisting of normalized positive linear functionals $\omega : E \to \mathbb{R}$ such that $\omega(1) = 1$. For any two states $\omega, \phi \in \mathcal{S}[\mathbb{R}, E]$, if $\omega(e) = \phi(e)$ for every effect $e \in E$ (where an effect is an element satisfying $0 \le e \le 1$), then $\omega = \phi$.

Given a unital positive linear map $\varphi : E_1 \to E_2$ and a state $\omega \in \mathcal{S}[\mathbb{R}, E_2]$, the pullback state is the state in $\mathcal{S}[\mathbb{R}, E_1]$ defined by the composition $\omega \circ \varphi$. In the context of quantum mechanics, this represents the Schrödinger-picture action of a Heisenberg-picture channel $\varphi$ on the state $\omega$, such that for any $x \in E_1$, the resulting state maps $x$ to $\omega(\varphi(x))$.

Let $E_1$ and $E_2$ be ordered real vector spaces with units. For any unital positive linear map $\varphi : E_1 \to E_2$, any state $\omega \in \mathcal{S}[\mathbb{R}, E_2]$, and any element $x \in E_1$, the value of the pullback state $\varphi.\text{pullbackState } \omega$ acting on $x$ is given by the application of $\omega$ to $\varphi(x)$: $$(\varphi.\text{pullbackState } \omega)(x) = \omega(\varphi(x))$$ This represents the Schrödinger-picture action of a Heisenberg-picture channel $\varphi$ on the state $\omega$.

Let $E_1$ be a unital ordered vector space over $\mathbb{R}$, and let $\mathcal{S}[\mathbb{R}, E_1]$ denote its state space. For any state $\omega \in \mathcal{S}[\mathbb{R}, E_1]$, the pullback of $\omega$ along the identity unital positive linear map $\text{id}_{E_1} : E_1 \to E_1$ is equal to $\omega$ itself.

Let $E_1$, $E_2$, and $E_3$ be ordered real vector spaces. Let $\varphi : E_1 \to E_2$ and $\psi : E_2 \to E_3$ be unital positive linear maps, and let $\omega \in \mathcal{S}[\mathbb{R}, E_3]$ be a state on $E_3$. The pullback of the state $\omega$ under the composition $\psi \circ \varphi$ is equal to the pullback of the state $\psi^* \omega$ under the map $\varphi$. That is, $$(\psi \circ \varphi)^* \omega = \varphi^* (\psi^* \omega)$$ where $(\cdot)^*$ denotes the pullback operation (the Schrödinger-picture action).