PhyslibAlpha.ProbabilisticTheory.State.Metric

For any state $\omega \in \mathcal{S}(\mathbb{R}, E)$ on an Archimedean order-unit space $E$ and any element $A \in E$, the value assigned by the state to $A$ is bounded above by the order-unit norm of $A$: $$\omega(A) \leq \|A\|$$ where $\|A\|$ denotes the order-unit norm of $A$, and $\omega(A)$ represents the expectation value of the observable $A$ in state $\omega$.

Let $E$ be an Archimedean order-unit space. For any state $\omega \in \mathcal{S}(\mathbb{R}, E)$ (defined as a unital positive linear functional) and any observable $A \in E$, the absolute value of the expectation value $\omega(A)$ is bounded by the order-unit norm $\|A\|_1$ of the observable: $$|\omega(A)| \leq \|A\|_1$$

For any two states $\omega, \phi \in \mathcal{S}(\mathbb{R}, E)$ on an Archimedean order-unit space $E$ and any observable $A \in E$ with order-unit norm $\|A\|_1 \leq 1$, the absolute difference between the values assigned by the states to $A$ is at most 2: $$|\omega(A) - \phi(A)| \leq 2$$ where $\omega(A)$ and $\phi(A)$ represent the "predictions" or expectation values of the observable $A$ in states $\omega$ and $\phi$ respectively.

Let $E$ be an Archimedean order-unit space and let $\omega$ and $\phi$ be two states (unital positive linear functionals) on $E$. The set of values $\{|\omega(A) - \phi(A)| : A \in E, \|A\|_1 \le 1\}$, where $\|A\|_1$ is the order-unit norm of $A$, is bounded above.

In an Archimedean order-unit space $E$, the set of elements $A \in E$ whose order-unit norm $\|A\|_1$ is less than or equal to $1$ is non-empty.

For two states $\omega, \phi \in \mathcal{S}[\mathbb{R}, E]$ on an Archimedean order-unit space $E$, the distance $\text{dist}(\omega, \phi)$ is defined as the operator-norm distance, which is the supremum of the absolute difference between their evaluations on any observable $A \in E$ with an order-unit norm $\|A\|_1 \le 1$: $$\text{dist}(\omega, \phi) = \sup \{ |\omega(A) - \phi(A)| : A \in E, \|A\|_1 \le 1 \}$$ This measures how far apart the predictions of the two states can be for an observable of norm at most $1$.

For any two states $\omega, \phi \in \mathcal{S}[\mathbb{R}, E]$ on an Archimedean order-unit space $E$, the distance between them is non-negative: $$\text{dist}(\omega, \phi) \ge 0$$ where $\text{dist}(\omega, \phi)$ is the operator-norm distance defined as the supremum of the absolute difference of their evaluations over elements of the space with order-unit norm at most $1$.

For any two states $\omega, \phi \in \mathcal{S}[\mathbb{R}, E]$ on an Archimedean order-unit space $E$, the operator-norm distance $\text{dist}(\omega, \phi)$ between them is at most $2$: $$\text{dist}(\omega, \phi) \le 2$$ This means that the diameter of the state space $\mathcal{S}[\mathbb{R}, E]$ is less than or equal to $2$.

For any state $\omega \in \mathcal{S}[\mathbb{R}, E]$ on an Archimedean order-unit space $E$, the distance between the state and itself is zero, i.e., $\text{dist}(\omega, \omega) = 0$.

For any two states $\omega, \phi \in \mathcal{S}[\mathbb{R}, E]$ on an Archimedean order-unit space $E$, the distance between them is symmetric: $$\text{dist}(\omega, \phi) = \text{dist}(\phi, \omega)$$

Let $E$ be an Archimedean order-unit space. For any three states $\omega, \phi, \psi \in \mathcal{S}[\mathbb{R}, E]$, the distance between them satisfies the triangle inequality: $$\text{dist}(\omega, \psi) \le \text{dist}(\omega, \phi) + \text{dist}(\phi, \psi)$$ where $\text{dist}$ is the operator-norm distance between states.

Let $E$ be an Archimedean order-unit space and let $\psi: E \to \mathbb{R}$ be a state (a unital positive linear map). For any effect $e \in E$ (an element such that $0 \le e \le 1$), let $2e - 1$ be the doubled and re-centered effect. The value of the state $\psi$ at this element satisfies $$\psi(2e - 1) = 2\psi(e) - 1.$$

The set of states $\mathcal{S}[\mathbb{R}, E]$ on an Archimedean order-unit space $E$ is a metric space. A state $\omega \in \mathcal{S}[\mathbb{R}, E]$ is a unital positive linear map from $E$ to $\mathbb{R}$. The metric is the operator-norm distance induced by the order-unit norm on $E$, where the distance between two states $\omega$ and $\phi$ is defined as: $$\text{dist}(\omega, \phi) = \sup \{ |\omega(A) - \phi(A)| : A \in E, \|A\| \le 1 \}$$ This structure equips the state space with the necessary properties of a metric space, including symmetry, the triangle inequality, and the identity of indiscernibles.

Let $E$ be an Archimedean order-unit space and $\mathcal{S}[\mathbb{R}, E]$ be the metric space of its states (unital positive linear maps $\omega: E \to \mathbb{R}$). For a given state $\omega$, this function calculates the infimum distance from $\omega$ to the set of pure states, defined as: $$\text{dist}(\omega, \text{Pure}) = \inf \{ \text{dist}(\omega, \phi) : \phi \in \mathcal{S}[\mathbb{R}, E], \phi \text{ is pure} \}$$ The distance is induced by the operator norm, and this value provides a measure of the "mixedness" of the state $\omega$.

For any state $\omega$ in the metric space of states $\mathcal{S}[\mathbb{R}, E]$ on an Archimedean order-unit space $E$, the distance to the set of pure states is non-negative: $0 \le \text{dist}(\omega, \text{Pure})$.

Let $E$ be an Archimedean order-unit space and $\mathcal{S}[\mathbb{R}, E]$ be the space of its states (unital positive linear maps $\omega: E \to \mathbb{R}$). If a state $\omega \in \mathcal{S}[\mathbb{R}, E]$ is a pure state, then its distance to the set of pure states is zero, i.e., $\text{dist}(\omega, \text{Pure}) = 0$.

Let $E$ be an Archimedean order-unit space and $\mathcal{S}[\mathbb{R}, E]$ be the metric space of its states (unital positive linear maps from $E$ to $\mathbb{R}$). The function $\text{distToPure} : \mathcal{S}[\mathbb{R}, E] \to \mathbb{R}$, which measures the infimum distance from a state $\omega$ to the set of pure states, is $1$-Lipschitz continuous. That is, for any two states $\omega, \phi \in \mathcal{S}[\mathbb{R}, E]$, the following inequality holds: $$|\text{distToPure}(\omega) - \text{distToPure}(\phi)| \le \text{dist}(\omega, \phi)$$ where $\text{dist}(\omega, \phi)$ is the operator-norm distance between the states.