PhyslibAlpha.QuantumMechanics.OperatorAlgebra.States.Basic

For a state $\omega$ on a $C^*$-algebra $A$, this definition provides the corresponding continuous linear map $\omega : A \to \mathbb{C}$. It leverages the fact that positive linear functionals on a $C^*$-algebra are automatically bounded, and thus continuous.

Let $A$ be an operator algebra and $\omega$ be a state on $A$. For any element $a \in A$, the value of the continuous linear map associated with $\omega$ applied to $a$ is equal to the value of the underlying positive linear functional applied to $a$.

Let $A$ be an operator algebra and $\omega$ be a state on $A$. When $\omega$ is viewed as a continuous linear map from $A$ to $\mathbb{C}$, the value of the state applied to the identity element $1 \in A$ is equal to $1$. This restates the normalization condition $\omega(1) = 1$ for the continuous linear map representation of the state.

Let $A$ be an operator algebra and $\omega$ be a state on $A$. For any element $a \in A$, the value of the state applied to the adjoint $a^*$ is the complex conjugate of the value of the state applied to $a$. That is, $\omega(a^*) = \overline{\omega(a)}$.

Let $A$ be an operator algebra and $\omega$ be a state on $A$. For any element $a \in A$, the expectation value of $a^* a$ is non-negative, i.e., $0 \le \omega(a^* a)$, where $a^*$ denotes the adjoint (or star) of $a$.

For any state $\omega$ on an operator algebra $A$ and any element $a \in A$, the absolute value of the expectation value $|\omega(a)|$ is less than or equal to the operator norm $\|a\|$ of the element: $$|\omega(a)| \le \|a\|$$

Let $A$ be an operator algebra and $\omega$ be a state on $A$. The operator norm of the state $\omega$, when considered as a continuous linear functional $\omega: A \to \mathbb{C}$, is exactly $1$: $$\|\omega\| = 1$$ This normalization property ensures that, physically, a state neither amplifies nor damps observables, as the expectation value satisfies $|\omega(a)| \le \|a\|$ with the bound being sharp.

Let $A$ be an operator algebra and $\omega$ be a state on $A$. For any self-adjoint element $a \in A$ (satisfying $a = a^*$), the imaginary part of the value $\omega(a)$ is zero, i.e., $\operatorname{Im}(\omega(a)) = 0$. In physical terms, the expectation value of a self-adjoint observable in any state is a real number.

This definition provides a coercion from the type of states $\text{State } A$ to the function type $A \to \mathbb{C}$. It allows a state $\omega$ to be applied directly to an element $a \in A$ to return its expectation value $\omega(a)$, which is defined as the result of the state's underlying positive linear functional.

Let $A$ be an operator algebra. For any two states $\omega$ and $\phi$ on $A$, if $\omega(a) = \phi(a)$ for every observable $a \in A$, then $\omega = \phi$. In this context, $\omega(a)$ denotes the expectation value of the observable $a$ in the state $\omega$.

Let $A$ be an operator algebra. The map that sends each state $\omega \in \text{State } A$ to its corresponding continuous linear functional $\omega : A \to \mathbb{C}$ is injective. That is, if two states induce the same continuous linear map on $A$, they are identical.