PhyslibAlpha.ProbabilisticTheory.Classical.NormalStates

Given a point $x$ in the sample space $\Omega$, the deterministic state at $x$ is defined as the unital positive linear map on the space of bounded measurable functions $\text{BoundedMeasurable}(\Omega)$ that maps each function $f$ to its value $f(x)$ at the point $x$. In other words, for an outcome $x \in \Omega$, it defines a state $\omega_x$ such that $\omega_x(f) = f(x)$.

For any outcome $x$ in the sample space $\Omega$ and any bounded measurable function $f \in \text{BoundedMeasurable}(\Omega)$, the deterministic state at $x$ (denoted by $\text{eval}_x$) applied to $f$ is equal to the value of the function at $x$, i.e., $\text{eval}_x(f) = f(x)$.

For any point $x$ in the sample space $\Omega$, the deterministic state $\text{eval}_x$ on the space of bounded measurable functions $\text{BoundedMeasurable}(\Omega)$, defined by $\text{eval}_x(f) = f(x)$, is normal. That is, for any non-empty directed subset $D \subseteq \text{BoundedMeasurable}(\Omega)$ with a supremum $f$, it holds that $f(x) = \sup_{g \in D} g(x)$.

For any outcome $x$ in a sample space $\Omega$, the deterministic state $\text{eval}_x$ (the unital positive linear map on the space of bounded measurable functions $\text{BoundedMeasurable}(\Omega)$ defined by $\text{eval}_x(f) = f(x)$) is a pure state. A state is pure if it is an extreme point of the convex set of states $\mathcal{S}(\text{BoundedMeasurable}(\Omega))$. In the context of these deterministic states, this property implies that the value of the smaller of two observables at $x$ is the smaller of their individual values.

Let $\Omega$ be a sample space and $\text{BoundedMeasurable}(\Omega)$ be the space of bounded measurable functions on $\Omega$. Let $\omega$ be a state (a unital positive linear map from $\text{BoundedMeasurable}(\Omega)$ to $\mathbb{R}$) on the system. For any observable $g \in \text{BoundedMeasurable}(\Omega)$ and any real number $\varepsilon$, if $g$ is pointwise bounded such that $|g(x)| \le \varepsilon$ for all $x \in \Omega$, then the expectation value of the observable under the state $\omega$ satisfies $|\omega(g)| \le \varepsilon$.

Given a probability measure $\mu$ on a sample space $\Omega$, this definition constructs a state on the space of bounded measurable functions $\text{BoundedMeasurable}(\Omega)$. The state maps each function $f \in \text{BoundedMeasurable}(\Omega)$ to its integral with respect to $\mu$: $$\omega(f) = \int_{\Omega} f(x) \, d\mu(x).$$ This map is a unital positive linear map, meaning it is linear, satisfies $\omega(f) \ge 0$ for $f \ge 0$, and preserves the identity $\omega(1) = 1$.

Let $\Omega$ be a sample space. For any probability measure $\mu$ on $\Omega$ and any bounded measurable function $f$ in the space $\text{BoundedMeasurable}(\Omega)$, the state associated with $\mu$ (denoted as $\text{ofMeasure}(\mu)$) evaluated at $f$ is equal to the integral of $f$ with respect to $\mu$: $$\text{ofMeasure}(\mu)(f) = \int_{\Omega} f(x) \, d\mu(x).$$

For any probability measure $\mu$ on a sample space $\Omega$, the state $\omega$ on the space of bounded measurable functions $\text{BoundedMeasurable}(\Omega)$ defined by integration against $\mu$, $$\omega(f) = \int_{\Omega} f(x) \, d\mu(x),$$ is a normal state. A state is normal if it preserves the suprema of directed sets: for any non-empty directed subset $D \subseteq \text{BoundedMeasurable}(\Omega)$ with a least upper bound $f$, it holds that $\omega(f) = \sup_{g \in D} \omega(g)$.

Let $\Omega$ be a measurable space and $\text{BoundedMeasurable}(\Omega)$ be the order-unit space of bounded measurable functions on $\Omega$. Given a normal state $\omega : \text{BoundedMeasurable}(\Omega) \to \mathbb{R}$ (a unital positive linear map that preserves suprema of directed sets), the **outcome distribution** is the probability measure $\mu$ on $\Omega$ defined by: $$\mu(s) = \omega(\mathbb{1}_s)$$ for every measurable set $s \subseteq \Omega$, where $\mathbb{1}_s$ is the indicator function of $s$. This construction identifies the normal states of a classical system with probability measures on its sample space.

Let $\Omega$ be a measurable space and $\text{BoundedMeasurable}(\Omega)$ be the space of bounded measurable functions on $\Omega$. Let $\omega$ be a normal state on $\text{BoundedMeasurable}(\Omega)$ (a unital positive linear map that preserves suprema of directed sets) and let $\mu$ be the probability measure on $\Omega$ associated with $\omega$. For any measurable set $s \subseteq \Omega$, the measure $\mu(s)$ is given by the value of the state $\omega$ applied to the indicator function $\mathbb{1}_s$: $$\mu(s) = \omega(\mathbb{1}_s)$$ where the real-valued result of the state is interpreted as an extended non-negative real number.

Let $\Omega$ be a measurable space and let $\omega$ be a normal state on the space of bounded measurable functions $\text{BoundedMeasurable}(\Omega)$. Let $\mu$ be the probability measure on $\Omega$ associated with $\omega$ (its outcome distribution). For any measurable set $s \subseteq \Omega$, the real-valued measure of $s$ satisfies $$\mu(s) = \omega(\mathbb{1}_s),$$ where $\mathbb{1}_s$ is the indicator function of the set $s$.

Let $\Omega$ be a measurable space and $\mu$ be a probability measure on $\Omega$. Let $\text{ofMeasure}(\mu)$ be the normal state on the space of bounded measurable functions $\text{BoundedMeasurable}(\Omega)$ defined by integration: $$\text{ofMeasure}(\mu)(f) = \int_{\Omega} f \, d\mu.$$ Then, the probability measure $\text{toMeasure}(\text{ofMeasure}(\mu))$ obtained from this state (by evaluating it on indicator functions of measurable sets) is equal to the original measure $\mu$.

Let $\Omega$ be a measurable space and $\text{BoundedMeasurable}(\Omega)$ be the space of bounded measurable functions on $\Omega$. Let $\omega$ be a normal state on this space (a unital positive linear map that preserves suprema of directed sets). Let $\mu_\omega$ be the probability measure (the outcome distribution) associated with $\omega$, defined such that for any measurable set $s$, $\mu_\omega(s) = \omega(\mathbb{1}_s)$. For any bounded measurable function $g$ and natural numbers $N$ and $K$, the value of the state $\omega$ on the staircase function $\text{staircase}(g, N, K)$ is equal to the integral of that staircase function with respect to $\mu_\omega$: $$\omega(\text{staircase}(g, N, K)) = \int_{\Omega} \text{staircase}(g, N, K)(x) \, d\mu_\omega(x).$$

Let $\Omega$ be a measurable space and $\text{BoundedMeasurable}(\Omega)$ be the space of bounded measurable functions from $\Omega$ to $\mathbb{R}$. Let $\omega$ be a normal state on $\text{BoundedMeasurable}(\Omega)$ and let $\mu$ be its associated probability measure (the outcome distribution) defined by $\mu(s) = \omega(\mathbb{1}_s)$ for every measurable set $s \subseteq \Omega$. For any non-negative bounded measurable function $g \in \text{BoundedMeasurable}(\Omega)$ that is bounded above by a natural number $K$ (i.e., $0 \le g(x) \le K$ for all $x \in \Omega$), the following identity holds: $$\int_{\Omega} g(x) \, d\mu(x) = \omega(g).$$

Let $\Omega$ be a measurable space and $\text{BoundedMeasurable}(\Omega)$ be the order-unit space of bounded measurable functions on $\Omega$. Let $\omega: \text{BoundedMeasurable}(\Omega) \to \mathbb{R}$ be a normal state. Let $\mu_\omega$ be the probability measure (outcome distribution) associated with $\omega$, defined by $\mu_\omega(s) = \omega(\mathbb{1}_s)$ for any measurable set $s \subseteq \Omega$, where $\mathbb{1}_s$ is the indicator function. Then the state $\omega$ is equal to the state defined by integration against $\mu_\omega$. That is, for any $f \in \text{BoundedMeasurable}(\Omega)$: $$\omega(f) = \int_{\Omega} f(x) \, d\mu_\omega(x).$$

Let $\Omega$ be a measurable space. There is an equivalence between the set of normal states on the space of bounded measurable functions $\text{BoundedMeasurable}(\Omega)$ and the set of probability measures on $\Omega$. A state $\omega \in \mathcal{S}[\mathbb{R}, \text{BoundedMeasurable}(\Omega)]$ (a unital positive linear map) is called **normal** if it preserves the suprema of directed sets. The equivalence is defined by the following maps: - To a normal state $\omega$, it assigns the probability measure $\mu$ defined by $\mu(s) = \omega(\mathbb{1}_s)$ for every measurable set $s \subseteq \Omega$, where $\mathbb{1}_s$ denotes the indicator function. - To a probability measure $\mu$, it assigns the normal state $\omega$ defined by integration: $\omega(f) = \int_{\Omega} f(x) \, d\mu(x)$ for every $f \in \text{BoundedMeasurable}(\Omega)$.