PhyslibAlpha.ProbabilisticTheory.State.Separation

Separation by states

i. Overview

In an Archimedean order-unit space, states determine the entire ordered normed structure. An element is positive exactly when every state assigns it a nonnegative value, and its order-unit norm is `‖A‖₁ = sup { |ω A| : ω a state }`. Consequently, states separate points: two elements are equal whenever every state assigns them the same value.

The main ingredient is Hahn–Banach separation of a point from the positive cone. Applied to an element `A ≱ 0`, it produces a state `ω` with `ω A < 0`. Everything else follows from this one separation fact.

ii. Key results

- `UnitalPositiveLinearMap.exists_apply_neg_of_not_nonneg` : every element outside the positive cone is separated from it by a state. - `UnitalPositiveLinearMap.nonneg_iff_forall_state_nonneg` : states determine the positive cone. - `UnitalPositiveLinearMap.sSup_abs_apply_eq_orderUnitNorm` : states determine the order-unit norm. - `UnitalPositiveLinearMap.ext_of_forall_apply_eq` : states separate points. - `UnitalPositiveLinearMap.toOrderUnitContinuousLinearMap` : a state as a continuous functional.

iii. Table of contents

A. Separating points from the cone
B. The positive cone
C. The order-unit norm
D. Separation of points
E. Continuous-dual realization

A. Separating points from the cone

B. The positive cone

C. The order-unit norm

D. Separation of points

E. Continuous-dual realization

13 declarations

theorem

$p(1) = 0 \implies p = 0$ for positive linear functionals

#apply_eq_zero_of_apply_one_eq_zero

Let $E$ be an Archimedean order-unit space with order unit $1$ . If $p: E \to \mathbb{R}$ is a positive linear functional such that $p(1) = 0$ , then $p(A) = 0$ for all $A \in E$ .

theorem

$A \not\ge 0 \implies \exists \text{ state } \omega, \omega(A) < 0$

#exists_apply_neg_of_not_nonneg

Let $E$ be an Archimedean order-unit space. If an element $A \in E$ is not in the positive cone (i.e., $A \not\ge 0$ ), then there exists a state $\omega \in \mathcal{S}(\mathbb{R}, E)$ such that $\omega(A) < 0$ . Here, a state is defined as a unital positive linear map from $E$ to $\mathbb{R}$ .

theorem

$0 \le A \iff \forall \omega \in \mathcal{S}(\mathbb{R}, E), 0 \le \omega(A)$

#nonneg_iff_forall_state_nonneg

Let $E$ be an Archimedean order-unit space. For any element $A \in E$ , $A$ is positive ( $0 \le A$ ) if and only if $\omega(A) \ge 0$ for every state $\omega$ in the state space $\mathcal{S}(\mathbb{R}, E)$ .

instance

Existence of a state in a nontrivial Archimedean order-unit space E

#instNonemptyState

Let $E$ be a nontrivial Archimedean order-unit space. The set of states on $E$ , denoted by $\mathcal{S}(\mathbb{R}, E)$ , is non-empty. In other words, every nontrivial Archimedean order-unit space possesses at least one state (a unital positive linear map from $E$ to $\mathbb{R}$ ).

theorem

If $A \not\leq r \cdot \mathbb{1}$ , then $\omega(A) > r$ for some state $\omega$

#exists_state_apply_gt_of_not_le

Let $E$ be an Archimedean order-unit space with order unit $\mathbb{1}$ . For any element $A \in E$ and any real number $r \in \mathbb{R}$ , if $A \not\leq r \cdot \mathbb{1}$ , then there exists a state $\omega$ (a unital positive linear map from $E$ to $\mathbb{R}$ ) such that $\omega(A) > r$ .

theorem

If $r < \|A\|_1$ , then $|\omega(A)| > r$ for some state $\omega$

#exists_state_abs_apply_gt_of_lt_orderUnitNorm

Let $E$ be a nontrivial Archimedean order-unit space. For any element $A \in E$ and any real number $r$ such that $r < \|A\|_1$ (where $\|\cdot\|_1$ denotes the order-unit norm), there exists a state $\omega$ (a unital positive linear map from $E$ to $\mathbb{R}$ ) such that $r < |\omega(A)|$ .

theorem

$\|A\|_{\mathbb{1}} = \sup_{\omega \in \mathcal{S}} |\omega(A)|$

#sSup_abs_apply_eq_orderUnitNorm

In a nontrivial Archimedean order-unit space $E$ , the order-unit norm of an element $A \in E$ , denoted by $\|A\|_{\mathbb{1}}$ , is equal to the supremum of the absolute values $|\omega(A)|$ over all states $\omega$ in the state space $\mathcal{S}(\mathbb{R}, E)$ : $\|A\|_{\mathbb{1}} = \sup_{\omega \in \mathcal{S}(\mathbb{R}, E)} |\omega(A)|$ where a state $\omega$ is a unital positive linear map from $E$ to $\mathbb{R}$ .

theorem

The norm of the order unit is $1$

#orderUnitNorm_one

In a nontrivial Archimedean order-unit space $E$ , the order-unit norm of the order unit $\mathbb{1} \in E$ is equal to $1$ , that is, $\|\mathbb{1}\| = 1$ .

theorem

States separate points in Archimedean order-unit spaces

#ext_of_forall_apply_eq

Let $E$ be an Archimedean order-unit space. The set of states $\mathcal{S}(\mathbb{R}, E)$ separates points in $E$ : for any elements $A, B \in E$ , if $\omega(A) = \omega(B)$ for every state $\omega \in \mathcal{S}(\mathbb{R}, E)$ , then $A = B$ .

definition

State $\omega$ as a continuous linear functional on $(E, \|\cdot\|_1)$

#toOrderUnitContinuousLinearMap

Let $E$ be an Archimedean order-unit space and let $\|\cdot\|_1$ denote its canonical order-unit norm. Given a state $\omega \in \mathcal{S}(\mathbb{R}, E)$ (which is a unital positive linear map from $E$ to $\mathbb{R}$ ), this definition constructs the corresponding continuous linear functional $\omega: (E, \|\cdot\|_1) \to \mathbb{R}$ . The continuity is derived from the property that for any state $\omega$ , the inequality $|\omega(x)| \le \|x\|_1$ holds for all $x \in E$ .

theorem

Evaluation of the continuous linear map associated with a state $\omega$ at $A$ equals $\omega(A)$

#toOrderUnitContinuousLinearMap_apply

Let $E$ be an Archimedean order-unit space. For any state $\omega \in \mathcal{S}(\mathbb{R}, E)$ and any element $A \in E$ , the value obtained by applying the continuous linear functional associated with $\omega$ to $A$ is equal to the original value $\omega(A)$ .

theorem

The mapping from states to continuous linear functionals is injective

#toOrderUnitContinuousLinearMap_injective

Let $E$ be an Archimedean order-unit space. Let $\mathcal{S}(\mathbb{R}, E)$ denote the set of states on $E$ , where a state is defined as a unital positive linear map from $E$ to $\mathbb{R}$ . For each state $\omega \in \mathcal{S}(\mathbb{R}, E)$ , let $\tilde{\omega} : (E, \|\cdot\|_{\mathbf{1}}) \to \mathbb{R}$ be the continuous linear functional corresponding to $\omega$ on the space $E$ equipped with its canonical order-unit norm $\|\cdot\|_{\mathbf{1}}$ . The mapping $\omega \mapsto \tilde{\omega}$ is injective.

theorem

The operator norm of a state is 1

#norm_toOrderUnitContinuousLinearMap

Let $E$ be a nontrivial Archimedean order-unit space equipped with its canonical order-unit norm $\|\cdot\|_{\mathbf{1}}$ . For any state $\omega$ on $E$ (viewed as a continuous linear functional $\omega: (E, \|\cdot\|_{\mathbf{1}}) \to \mathbb{R}$ ), the operator norm of $\omega$ is equal to 1.

PhyslibAlpha.ProbabilisticTheory.State.Separation

Separation by states

i. Overview

ii. Key results

iii. Table of contents

A. Separating points from the cone

B. The positive cone

C. The order-unit norm

D. Separation of points

E. Continuous-dual realization

p(1)=0 ⟹ p=0p(1) = 0 \implies p = 0p(1)=0⟹p=0 for positive linear functionals

A≱0 ⟹ ∃ state ω,ω(A)<0A \not\ge 0 \implies \exists \text{ state } \omega, \omega(A) < 0A≥0⟹∃ state ω,ω(A)<0

0≤A ⟺ ∀ω∈S(R,E),0≤ω(A)0 \le A \iff \forall \omega \in \mathcal{S}(\mathbb{R}, E), 0 \le \omega(A)0≤A⟺∀ω∈S(R,E),0≤ω(A)

Existence of a state in a nontrivial Archimedean order-unit space E

If A≰r⋅1A \not\leq r \cdot \mathbb{1}A≤r⋅1, then ω(A)>r\omega(A) > rω(A)>r for some state ω\omegaω

If r<∥A∥1r < \|A\|_1r<∥A∥1​, then ∣ω(A)∣>r|\omega(A)| > r∣ω(A)∣>r for some state ω\omegaω

∥A∥1=sup⁡ω∈S∣ω(A)∣\|A\|_{\mathbb{1}} = \sup_{\omega \in \mathcal{S}} |\omega(A)|∥A∥1​=supω∈S​∣ω(A)∣

The norm of the order unit is 111

States separate points in Archimedean order-unit spaces

State ω\omegaω as a continuous linear functional on (E,∥⋅∥1)(E, \|\cdot\|_1)(E,∥⋅∥1​)

Evaluation of the continuous linear map associated with a state ω\omegaω at AAA equals ω(A)\omega(A)ω(A)

The mapping from states to continuous linear functionals is injective

The operator norm of a state is 1

$p(1) = 0 \implies p = 0$ for positive linear functionals

$A \not\ge 0 \implies \exists \text{ state } \omega, \omega(A) < 0$

$0 \le A \iff \forall \omega \in \mathcal{S}(\mathbb{R}, E), 0 \le \omega(A)$

If $A \not\leq r \cdot \mathbb{1}$ , then $\omega(A) > r$ for some state $\omega$

If $r < \|A\|_1$ , then $|\omega(A)| > r$ for some state $\omega$

$\|A\|_{\mathbb{1}} = \sup_{\omega \in \mathcal{S}} |\omega(A)|$

The norm of the order unit is $1$

State $\omega$ as a continuous linear functional on $(E, \|\cdot\|_1)$

Evaluation of the continuous linear map associated with a state $\omega$ at $A$ equals $\omega(A)$