PhyslibAlpha.ProbabilisticTheory.Classical.LatticeObservables

For a unital positive linear map $\omega: E \to \mathbb{R}$ and a fixed element $u \in E$, the band component maps an element $h \in E$ to the supremum $\sup_{a \ge 0} \omega(h \wedge au)$, where $\wedge$ denotes the lattice infimum. This value represents the part of the functional $\omega$ that is "carried" by the element $u$.

Let $E$ be an order-unit lattice with order unit $u$, and let $\omega: E \to \mathbb{R}$ be a unital positive linear map. For any element $h \in E$, the set of expectation values $\{ \omega(h \sqcap au) : a \in \mathbb{R}_{\ge 0} \}$ is bounded above, where $\sqcap$ denotes the infimum in the lattice.

Let $E$ be an order-unit lattice with order unit $u$, and let $\omega: E \to \mathbb{R}$ be a unital positive linear map. For any element $h \in E$ and any non-negative real number $a \ge 0$, the expectation value of the lattice infimum $\omega(h \sqcap au)$ is less than or equal to the band component $\text{band}(\omega, u, h)$.

Let $E$ be an order-unit lattice and $\omega: E \to \mathbb{R}$ be a unital positive linear map. For any element $h \in E$ and any real number $r$, if the expectation value $\omega(h \sqcap au)$ is less than or equal to $r$ for all non-negative reals $a \geq 0$, then the band component $\text{band}(\omega, u, h)$ is also less than or equal to $r$.

Let $E$ be an order-unit lattice with order unit $u$, and let $\omega : E \to \mathbb{R}$ be a unital positive linear map. For any non-negative element $h \in E$ (such that $h \ge 0$), its band component $\sup_{a \ge 0} \omega(h \wedge au)$ is non-negative.

Let $E$ be an order-unit lattice and $\omega: E \to \mathbb{R}$ be a unital positive linear map. Given a non-negative element $u \in E$ ($u \ge 0$), for any non-negative elements $h_1, h_2 \in E$, the band component $\text{band}(\omega, u, \cdot)$ is additive: $$\text{band}(\omega, u, h_1 + h_2) = \text{band}(\omega, u, h_1) + \text{band}(\omega, u, h_2)$$ where the band component is defined as $\text{band}(\omega, u, h) = \sup_{a \ge 0} \omega(h \wedge au)$, and $\wedge$ denotes the lattice infimum.

Let $E$ be an order-unit lattice with order unit $u$, and let $\omega : E \to \mathbb{R}$ be a unital positive linear map. For any non-negative real number $c \ge 0$, any $h \in E$, and any $a \in \mathbb{R}$, the following equality holds: $$\omega(c \cdot h \sqcap (ca) \cdot u) = c \cdot \omega(h \sqcap a \cdot u)$$ where $\sqcap$ denotes the infimum (meet) operator in the lattice $E$.

Let $E$ be an order-unit lattice and $\omega: E \to \mathbb{R}$ be a unital positive linear map. For a fixed element $u \in E$, the band component of an element $h \in E$ is defined as $\text{band}(\omega, u, h) = \sup_{a \ge 0} \omega(h \wedge au)$, where $\wedge$ denotes the lattice infimum. For any element $h \in E$ and any strictly positive real number $c > 0$, the band component satisfies the following homogeneity property: $$\text{band}(\omega, u, c \cdot h) = c \cdot \text{band}(\omega, u, h)$$

Let $E$ be an order-unit lattice and $\omega: E \to \mathbb{R}$ be a unital positive linear map. Given a non-negative element $u \in E$ ($u \ge 0$), the band map is a positive linear map from $E$ to $\mathbb{R}$ representing the part of the functional $\omega$ that is "carried" by $u$. For any non-negative element $h \in E$, the value of the map is defined by the supremum: $$\sup_{a \ge 0} \omega(h \wedge au)$$ where $\wedge$ denotes the lattice infimum (meet) operator.

Let $E$ be an order-unit lattice and $\omega: E \to \mathbb{R}$ be a unital positive linear map. For any non-negative elements $u \in E$ and $h \in E$ (i.e., $u \ge 0$ and $h \ge 0$), the value of the positive linear map $\text{bandMap}(\omega, u)$ at $h$ is equal to the band component $\text{band}(\omega, u, h)$, defined as: $$\text{bandMap}(\omega, u)(h) = \text{band}(\omega, u, h) = \sup_{a \ge 0} \omega(h \wedge au)$$ where $\wedge$ denotes the lattice infimum.

Let $E$ be an order-unit lattice and let $\omega : E \to \mathbb{R}$ be a unital positive linear map. For any non-negative observable $u \in E$ ($u \ge 0$), the band map $\text{band}(\omega, u, \cdot)$ is less than or equal to the original functional $\omega$. That is, for any $h \in E$ with $h \ge 0$, the value of the band map satisfies $\text{band}(\omega, u, h) \le \omega(h)$.

Let $E$ be an order-unit lattice and $\omega: E \to \mathbb{R}$ be a unital positive linear map. For any non-negative element $u \in E$ ($u \ge 0$), let $\text{band}(\omega, u, \cdot)$ be the band map associated with $u$. Then the value of this band map at $u$ is equal to the value of the original map at $u$: $$\text{band}(\omega, u, u) = \omega(u)$$

Let $E$ be an order-unit lattice and $\omega: E \to \mathbb{R}$ be a unital positive linear map. For any non-negative elements $u, g \in E$ such that $u$ and $g$ are disjoint (i.e., their lattice infimum satisfies $g \wedge u = 0$), the band map of $\omega$ associated with $u$ vanishes on $g$: $$\text{band}(\omega, u, g) = 0$$

Let $E$ be a vector lattice and $\omega$ be a pure state on $E$. For any two positive elements $g, u \in E$ such that they are disjoint (i.e., $g \wedge u = 0$), then either $\omega(g) = 0$ or $\omega(u) = 0$.

Let $E$ be an order-unit lattice and $\omega$ be a pure state in the state space $\mathcal{S}(E)$. For any two observables $f, g \in E$, the state $\omega$ preserves the infimum, such that the expectation value of the infimum of $f$ and $g$ is the minimum of their individual expectation values: $$\omega(f \sqcap g) = \min(\omega(f), \omega(g))$$

Let $E$ be an order-unit lattice and $\omega$ be a state (a unital positive linear map) on $E$. If $\omega$ preserves infima, such that $\omega(f \sqcap g) = \min(\omega(f), \omega(g))$ for all observables $f, g \in E$, then for any observable $f \in E$, the expectation of the absolute deviation of $f$ from its expectation value $\omega(f)$ is zero: $$\omega(|f - \omega(f) \cdot \mathbf{1}|) = 0,$$ where $\mathbf{1}$ denotes the order unit of $E$ and $| \cdot |$ denotes the lattice modulus.

Let $E$ be a space of lattice-ordered observables and $\omega: E \to \mathbb{R}$ be a state, represented as a unital positive linear map. If $\omega$ preserves infima such that $\omega(f \sqcap g) = \min(\omega(f), \omega(g))$ for all $f, g \in E$, then $\omega$ is a pure state.

Let $E$ be an order-unit lattice and $\omega \in \mathcal{S}(E)$ be a state (a unital positive linear map from $E$ to $\mathbb{R}$). The state $\omega$ is pure if and only if it preserves infima, meaning that for all observables $f, g \in E$, the condition $\omega(f \sqcap g) = \min(\omega(f), \omega(g))$ holds.

Let $E$ be an Archimedean order-unit lattice. The state space $\mathcal{S}(E)$ is the set of normalized positive continuous linear functionals in the weak-star dual space $E^*$. The subset of pure states $\{\omega \in \mathcal{S}(E) \mid \omega \text{ is pure}\}$ is a closed set in the weak-star topology.

Let $E$ be an Archimedean order-unit lattice. The space of pure states of $E$, denoted as $\text{PureState}(E)$ and equipped with the weak-* topology, is a compact topological space. Here, a pure state is an extreme point of the state space, which consists of all normalized positive continuous linear functionals on $E$.

Let $E$ be an Archimedean order-unit lattice. For any two observables $f, g \in E$, the evaluation of their infimum $f \sqcap g$ on the space of pure states is equal to the pointwise infimum of their individual evaluations. That is, for every pure state $\omega$, the expectation value of the infimum is the minimum of the expectation values of $f$ and $g$: $$\omega(f \sqcap g) = \min(\omega(f), \omega(g)).$$

In an Archimedean order-unit lattice $E$, for any observables $f, g \in E$, the evaluation of their supremum $f \sqcup g$ on the space of pure states is equal to the pointwise supremum of their individual evaluations. That is, for any pure state $\omega$, $\omega(f \sqcup g) = \max(\omega(f), \omega(g))$.

Let $E$ be an Archimedean order-unit lattice and let $\mathcal{P}(E)$ be the set of pure states of $E$. For each element $f \in E$, let $\hat{f} : \mathcal{P}(E) \to \mathbb{R}$ be the evaluation function defined by $\hat{f}(\omega) = \omega(f)$. The set of functions $\{\hat{f} \mid f \in E\}$ strongly separates the points of $\mathcal{P}(E)$. That is, for any two distinct pure states $\omega_1, \omega_2 \in \mathcal{P}(E)$, there exists an element $f \in E$ such that $\omega_1(f) \neq \omega_2(f)$, and for every pure state $\omega \in \mathcal{P}(E)$, there exists an element $f \in E$ such that $\omega(f) \neq 0$.

Let $E$ be an Archimedean order-unit lattice. Let $P(E)$ denote the set of pure states of $E$ (the extreme points of the state space). For every element $f \in E$, define the evaluation function $\hat{f}: P(E) \to \mathbb{R}$ by $\hat{f}(\omega) = \omega(f)$ for all $\omega \in P(E)$. The set of all such functions $\{\hat{f} \mid f \in E\}$ is dense in the space of continuous real-valued functions $C(P(E), \mathbb{R})$ with respect to the supremum norm.

Let $E$ be an Archimedean order-unit lattice. The state space $\mathcal{S}(E)$ is a Bauer simplex. This means that: 1. $\mathcal{S}(E)$ is a simplex, where every state $\omega \in \mathcal{S}(E)$ is uniquely represented by a regular probability measure on the set of pure states. 2. The set of pure states $\{\omega \in \mathcal{S}(E) \mid \omega \text{ is pure}\}$ is a closed set in the weak-* topology.

Let $E$ be an Archimedean order-unit lattice. For every state $\phi$ of $E$ (that is, a unital positive linear map $\phi: E \to \mathbb{R}$), there exists a unique regular probability measure $\mu$ supported on the set of pure states $P(E)$ (the extreme points of the state space) that represents $\phi$. This means that for every observable $f \in E$, the expectation value $\phi(f)$ is given by the integral: $$\phi(f) = \int_{P(E)} \omega(f) \, d\mu(\omega)$$

Let $\Omega$ be a measurable space and let $B(\Omega)$ be the space of bounded measurable real-valued functions on $\Omega$. The state space $\mathcal{S}(B(\Omega))$, consisting of all unital positive linear functionals on $B(\Omega)$, is a Bauer simplex. This implies that: 1. The set of pure states (the extreme points of the state space) is closed in the weak-* topology. 2. Every state $\omega \in \mathcal{S}(B(\Omega))$ is uniquely represented by a regular probability measure supported on the set of pure states.

Let $\Omega$ be a measurable space and let $B(\Omega)$ be the space of bounded measurable functions on $\Omega$. For any state $\phi \in \mathcal{S}(\mathbb{R}, B(\Omega))$, there exists a unique regular probability measure $\mu$ supported on the set of pure states of $B(\Omega)$ that represents $\phi$. This means that for every observable $f \in B(\Omega)$, the expectation value is given by the integral $\phi(f) = \int \psi(f) \, d\mu(\psi)$, where the integral is taken over the pure states $\psi$.

The space of observables in a finite classical system over an index set $\iota$, represented by the function space $\iota \to \mathbb{R}$, is equipped with the structure of an Archimedean order-unit lattice. This structure integrates the properties of an Archimedean order-unit space with the pointwise lattice operations, where for any two observables $f, g: \iota \to \mathbb{R}$, their infimum $f \wedge g$ and supremum $f \vee g$ are defined by $(f \wedge g)(i) = \min(f(i), g(i))$ and $(f \vee g)(i) = \max(f(i), g(i))$ for all $i \in \iota$.

For a finite classical system with observables represented by the function space $\mathbb{R}^\iota$ over a finite index set $\iota$, the state space $\mathcal{S}(\mathbb{R}^\iota)$ is a Bauer simplex. This implies that the set of pure states is closed in the weak-* topology and every state $\omega \in \mathcal{S}(\mathbb{R}^\iota)$ is uniquely represented by a regular probability measure supported on the set of pure states.