PhyslibAlpha.ProbabilisticTheory.State.Basic

States

i. Overview

A state assigns each observable its expectation value: a positive linear functional (`0 ≤ A ⟹ 0 ≤ ω A`), normalized so the certain outcome reads `ω 1 = 1`. In quantum mechanics, `ω A = tr(ρA)` for a density matrix `ρ`.

ii. Key results

`𝓢[𝕜, A]` : states on an ordered `𝕜`-vector space with a distinguished unit.

iii. Table of contents

A. Notation for positive functionals and states

A. Notation for positive functionals and states

2 declarations

definition

Positive linear functionals $\mathcal{P}[\mathbf{k}, A]$ on an ordered $\mathbf{k}$ -vector space

#term𝓟[_,_]

Given an ordered $\mathbf{k}$ -vector space $A$ over a scalar field $\mathbf{k}$ , $\mathcal{P}[\mathbf{k}, A]$ denotes the space of all positive linear functionals from $A$ to $\mathbf{k}$ . A linear functional $\omega: A \to \mathbf{k}$ is considered positive if $\omega(a) \geq 0$ for all $a \in A$ such that $a \geq 0$ .

definition

Space of states $\mathcal{S}[\mathbb{k}, A]$ on an ordered vector space $A$

#term𝓢[_,_]

For an ordered $\mathbb{k}$ -vector space $A$ with a distinguished unit $1_A$ , the notation $\mathcal{S}[\mathbb{k}, A]$ denotes the set of states on $A$ . A state is defined as a positive linear functional $\omega: A \to \mathbb{k}$ such that $\omega(a) \geq 0$ for all $a \ge 0$ and $\omega(1_A) = 1$ .