PhyslibAlpha.ProbabilisticTheory.Classical.UniqueDecomposition

Let $e$ be a pure state in the state space $\mathcal{S}(E)$ of a real vector space $E$. Let $\{s_i\}_{i \in I}$ be a family of elements in $E$ and $M$ be a real number. If the expectation value of $e$ on every element of the family is bounded by $M$ (i.e., $e(s_i) \leq M$ for all $i \in I$), then the upper envelope of the family $s$ evaluated at $e$ is also at most $M$.

Given an Archimedean order-unit space $E$ and a finite family of observables $s = \{A_i\}_{i \in I}$ in $E$, the function maps each state $\omega$ in the state space $\mathcal{S}(E)$ to the maximum expectation value $\max_{i \in I} \omega(A_i)$. The resulting object is a continuous real-valued function on the state space, denoted by $C(\mathcal{S}(E), \mathbb{R})$.

Let $E$ be an Archimedean order-unit space and $\{A_i\}_{i \in I}$ be a finite family of observables in $E$. For any state $\omega$ in the state space $\mathcal{S}(E)$, the maximum expectation value function evaluated at $\omega$ is equal to the maximum of the expectation values of the family, i.e., $$\text{maxApply}(\{A_i\})(\omega) = \max_{i \in I} \omega(A_i)$$ where $\omega(A_i)$ denotes the value of the state $\omega$ (viewed as a unital positive linear map) applied to the observable $A_i$.

Let $E$ be an Archimedean order-unit space and $\mathcal{S}(E)$ its state space. For any finite family of observables $\{A_i\}_{i \in I}$ in $E$ and any state $\omega \in \mathcal{S}(E)$, the expectation value of an individual observable $A_i$ is bounded above by the maximum expectation value of the family: $$\omega(A_i) \le \max_{j \in I} \omega(A_j).$$

Let $E$ be an Archimedean order-unit space and $\mathcal{S}(E)$ be its state space. Let $\{A_i\}_{i \in I}$ be a finite family of observables in $E$. For any state $\omega \in \mathcal{S}(E)$ and any real number $r$, if the expectation value of each observable satisfies $\omega(A_i) \le r$ for all $i \in I$, then the maximum expectation value $\max_{i \in I} \omega(A_i)$ also satisfies $\max_{i \in I} \omega(A_i) \le r$.

Let $E$ be an Archimedean order-unit space and $\mathcal{S}(E)$ its state space. Given a finite family of observables $\{A_i\}_{i \in I}$ in $E$, then for any state $\omega \in \mathcal{S}(E)$, there exists an index $j \in I$ such that the maximum expectation value of the family is attained at $A_j$, i.e., $$\max_{i \in I} \omega(A_i) = \omega(A_j).$$

Let $E$ be an Archimedean order-unit space and let $\{A_i\}_{i \in I}$ be a finite family of observables in $E$. If $b \in E$ is an upper bound of this family (i.e., $A_i \le b$ for all $i \in I$), then for any state $\omega$ in the state space $\mathcal{S}(E)$, the maximum expectation value of the family satisfies: $$\max_{i \in I} \omega(A_i) \le \omega(b)$$

Let $E$ be an Archimedean order-unit space and $\mathcal{S}(E)$ its state space. For any finite family of elements $\{A_i\}_{i \in I}$ in $E$ and any state $\omega \in \mathcal{S}(E)$, the absolute value of the maximum expectation value satisfies: $$\left| \max_{i \in I} \omega(A_i) \right| \le \sum_{i \in I} \|A_i\|_1$$ where $\|\cdot\|_1$ denotes the order-unit norm.

Let $E$ be an Archimedean order-unit space and $\mathcal{S}(E)$ be its state space. For any observable $a \in E$ and any state $\omega \in \mathcal{S}(E)$, the maximum expectation value of the singleton family $\{a\}$ evaluated at $\omega$ is equal to the expectation value $\omega(a)$.

Let $E$ be an Archimedean order-unit space with state space $\mathcal{S}(E)$. For any two finite families of observables $s: \iota \to E$ and $t: \kappa \to E$, let $\text{maxApply } s$ be the continuous function on $\mathcal{S}(E)$ defined by $\omega \mapsto \max_{i \in \iota} \omega(s_i)$. Then the maximum expectation value function for the combined family $s \oplus t$ (the disjoint union of the two families) is the pointwise supremum of the individual functions: $$\text{maxApply}(s \oplus t) = (\text{maxApply } s) \sqcup (\text{maxApply } t)$$ where $\sqcup$ denotes the pointwise maximum in the space of continuous functions $C(\mathcal{S}(E), \mathbb{R})$.

Let $E$ be an Archimedean order-unit space with state space $\mathcal{S}(E)$. Given two finite families of observables $s: I \to E$ and $t: J \to E$, let $\text{maxApply}_s$ and $\text{maxApply}_t$ be the continuous functions on the state space defined by $\omega \mapsto \max_{i \in I} \omega(s_i)$ and $\omega \mapsto \max_{j \in J} \omega(t_j)$, respectively. Then the function corresponding to the family of sums $\{s_i + t_j\}_{(i,j) \in I \times J}$ is equal to the sum of these two functions. That is, for every state $\omega \in \mathcal{S}(E)$, $$\max_{(i, j) \in I \times J} \omega(s_i + t_j) = \max_{i \in I} \omega(s_i) + \max_{j \in J} \omega(t_j).$$

Let $E$ be an Archimedean order-unit space and let $\{A_i\}_{i \in I}$ be a finite family of observables in $E$. For any measure $\mu$ defined on the set of pure states of $E$, the function that maps each pure state $\omega$ to the maximum expectation value $\max_{i \in I} \omega(A_i)$ is integrable with respect to $\mu$.

Let $E$ be an Archimedean order-unit space and $s = \{A_i\}_{i \in I}$ be a finite family of observables in $E$. Let $\mu$ be a finite measure on the set of pure states $\text{PureState}(E)$, and let $\phi: E \to \mathbb{R}$ be the positive linear functional defined by $\phi(A) = \int_{\text{PureState}(E)} \omega(A) \, d\mu(\omega)$. Then the integral of the maximum expectation value $\omega \mapsto \max_{i \in I} \omega(A_i)$ over the pure states is bounded above by the upper envelope of the family $s$ evaluated at $\phi$: $$\int_{\text{PureState}(E)} \max_{i \in I} \omega(A_i) \, d\mu(\omega) \le \widehat{s}(\phi)$$ where $\widehat{s}(\phi) = \inf \{ \phi(B) \mid B \in E, \forall \omega \in \text{StateSpace}(E), \omega(B) \ge \max_{i \in I} \omega(A_i) \}$.

Let $E$ be an Archimedean order-unit space and let $\mu$ be a finite measure on the set of pure states $\text{PureState}(E)$. Let $L_\mu: E \to \mathbb{R}$ be the positive linear functional defined by integrating elements of $E$ against $\mu$, i.e., $L_\mu(f) = \int_{\text{PureState}(E)} \omega(f) \, d\mu(\omega)$. The restriction of $\mu$ to the universal set of pure states, denoted $\mu|_{\text{univ}}$, yields the same positive linear functional as $\mu$ itself: $$L_{\mu|_{\text{univ}}} = L_\mu.$$

Let $E$ be an Archimedean order-unit space and let $\mu$ be a measure on its set of pure states $\text{PureState}(E)$. For any subset $A \subseteq \text{PureState}(E)$ and any measurable set $W \subseteq \text{PureState}(E)$, the positive linear functional $L_{\mu|_S}$ induced by the restriction of the measure $\mu$ to a set $S$ satisfies: $$L_{\mu|_{A \cap W}} + L_{\mu|_{A \setminus W}} = L_{\mu|_A}$$ where the induced functional $L_\nu: E \to \mathbb{R}$ for a measure $\nu$ is defined by $L_\nu(f) = \int_{\text{PureState}(E)} \omega(f) \, d\nu(\omega)$.

Let $E$ be an Archimedean order-unit space with order unit $\mathbf{1}$ and order-unit norm $\|\cdot\|_1$. For any finite family of elements $\{s_i\}_{i \in \iota}$ in $E$, the element $\left( \sum_{i \in \iota} \|s_i\|_1 \right) \mathbf{1}$ is an upper bound for the set $\{s_i\}_{i \in \iota}$. That is, for every $j \in \iota$, $$s_j \le \left( \sum_{i \in \iota} \|s_i\|_1 \right) \mathbf{1}.$$

Let $E$ be an Archimedean order-unit space and $\{s_i\}_{i \in \iota}$ be a finite family of elements in $E$. Let $\mu$ be a finite measure on the set of pure states $\text{PureState}(E)$. For any subset $A \subseteq \text{PureState}(E)$, let $L_A$ be the positive linear functional defined by integrating observables over $A$ against $\mu$, i.e., $L_A(f) = \int_A \omega(f) \, d\mu(\omega)$. Then the upper envelope $\hat{s}$ of the family evaluated at $L_A$ satisfies: $$\hat{s}(L_A) \le \int_A \max_{i \in \iota} \omega(s_i) \, d\mu(\omega) + 2 \left( \sum_{i \in \iota} \|s_i\|_1 \right) \mu(A)$$ where $\|s_i\|_1$ denotes the order-unit norm of $s_i$ and $\mu(A)$ is the measure of the set $A$. In other words, on any piece $A$, the upper envelope exceeds the integral of the largest expectation value by at most twice the sum of the norms of the family times the mass of the piece.

Let $E$ be an Archimedean order-unit space and $s = \{A_i\}_{i \in I}$ be a finite family of observables in $E$. Let $\mu$ be a finite measure on the set of pure states $\text{PureState}(E)$. For any pure state $e \in \text{PureState}(E)$ and any $\epsilon > 0$, there exists an open neighborhood $W$ of $e$ such that for every measurable subset $A \subseteq W$, the upper envelope $\widehat{s}$ evaluated on the positive functional $L_A(f) = \int_A \omega(f) \, d\mu(\omega)$ satisfies the inequality: $$\widehat{s}(L_A) \le \int_A \max_{i \in I} \omega(A_i) \, d\mu(\omega) + \epsilon \mu(A),$$ where $\mu(A)$ is the measure of the set $A$ and $\max_{i \in I} \omega(A_i)$ is the maximum expectation value of the family $s$ at the state $\omega$.

Let $E$ be an Archimedean order-unit space and $\iota$ be a non-empty index set. For any finite family of observables $s = \{A_i\}_{i \in \iota}$ in $E$ and any finite measure $\mu$ on the set of pure states $\text{PureState}(E)$, let $L$ be the positive linear functional on $E$ induced by the restriction of $\mu$ to the empty set (which is the zero functional). Then the upper envelope of the family $s$ evaluated at $L$ satisfies $\widehat{s}(L) \le 0$.

Let $E$ be an Archimedean order-unit space and $s = \{A_i\}_{i \in I}$ a finite family of observables in $E$. Let $\mu$ be a finite measure on the set of pure states $\text{PureState}(E)$. For any subset $S \subseteq \text{PureState}(E)$, let $\phi(S)$ denote the positive linear functional on $E$ defined by $\phi(S)(B) = \int_S \omega(B) \, d\mu(\omega)$. Given a subset $A \subseteq \text{PureState}(E)$ and a measurable subset $W \subseteq \text{PureState}(E)$, if there exist constants $r_1, r_2 \in \mathbb{R}$ such that: 1. $\text{upperEnvelope}_s(\phi(A \cap W)) \le \int_{A \cap W} \max_{i \in I} \omega(A_i) \, d\mu(\omega) + r_1$ 2. $\text{upperEnvelope}_s(\phi(A \setminus W)) \le \int_{A \setminus W} \max_{i \in I} \omega(A_i) \, d\mu(\omega) + r_2$ then it holds that: $$\text{upperEnvelope}_s(\phi(A)) \le \int_A \max_{i \in I} \omega(A_i) \, d\mu(\omega) + (r_1 + r_2)$$

Let $E$ be an Archimedean order-unit space and $s = \{A_i\}_{i \in I}$ be a finite family of observables in $E$. Let $\mu$ be a finite measure on the set of pure states $\text{PureState}(E)$. For any measurable set $A \subseteq \text{PureState}(E)$, let $\alpha_A$ be the positive linear functional on $E$ defined by the integral: $$\alpha_A(f) = \int_A \omega(f) \, d\mu(\omega)$$ Suppose there exists a collection of measurable sets $\{W_e\}_{e \in \text{PureState}(E)}$ and a constant $\epsilon \in \mathbb{R}$ such that for every $e$, any measurable subset $A \subseteq W_e$ satisfies the inequality: $$\hat{s}(\alpha_A) \leq \int_A \max_{i \in I} \omega(A_i) \, d\mu(\omega) + \epsilon \mu(A)$$ where $\hat{s}$ denotes the upper envelope of the family $s$ (the evaluation of the state at the supremum of $s$ in the state space). Then, for any finite collection of indices $t \subset \text{PureState}(E)$, if a measurable set $A$ is contained in the union $\bigcup_{e \in t} W_e$, the same inequality holds: $$\hat{s}(\alpha_A) \leq \int_A \max_{i \in I} \omega(A_i) \, d\mu(\omega) + \epsilon \mu(A)$$

Let $E$ be an Archimedean order-unit space and $s = \{A_i\}_{i \in I}$ be a finite family of observables in $E$. Let $\text{PureState}(E)$ denote the set of pure states of $E$. For any finite measure $\mu$ on $\text{PureState}(E)$, any compact subset $C \subseteq \text{PureState}(E)$, and any positive real number $\epsilon > 0$, the upper envelope of the family $s$ evaluated at the positive linear functional $L_C$ (where $L_C(A) = \int_C \omega(A) \, d\mu(\omega)$) satisfies the inequality: $$\text{upperEnvelope}(s, L_C) \le \int_C \max_{i \in I} \omega(A_i) \, d\mu(\omega) + \epsilon \cdot \mu(C)$$ where $\mu(C)$ is the measure of the set $C$.

Let $E$ be an Archimedean order-unit space and let $\text{PureState } E$ denote the set of its pure states. Given a finite regular measure $\mu$ on $\text{PureState } E$ and any positive real number $\epsilon > 0$, there exists a compact subset $C \subseteq \text{PureState } E$ such that the measure of the complement of $C$ is less than $\epsilon$, i.e., $\mu(\text{univ} \setminus C) < \epsilon$.

Let $E$ be an Archimedean order-unit space and let $\text{PureState}(E)$ denote the set of its pure states. Let $s = \{A_i\}_{i \in I}$ be a finite family of observables in $E$. Suppose $\mu$ is a regular probability measure on $\text{PureState}(E)$, and let $L_\mu: E \to \mathbb{R}$ be the positive linear functional defined by $L_\mu(A) = \int_{\text{PureState}(E)} \omega(A) \, d\mu(\omega)$. For any $\epsilon > 0$, the upper envelope of $s$ at $L_\mu$ is bounded as follows: $$\text{upperEnvelope}(s, L_\mu) \le \int_{\text{PureState}(E)} \max_{i \in I} \omega(A_i) \, d\mu(\omega) + \epsilon \left( 1 + 2 \sum_{i \in I} \|A_i\|_1 \right)$$ where $\|A_i\|_1$ is the order-unit norm of $A_i$.

Let $E$ be an Archimedean order-unit space satisfying the ensemble refinement property. Let $\mu$ be a regular probability measure on the set of pure states $\text{PureState}(E)$ of $E$, and let $\phi \in E^*$ be its barycenter, defined by the positive linear functional $\phi(A) = \int_{\text{PureState}(E)} \omega(A) \, d\mu(\omega)$ for all $A \in E$. For any finite family of observables $s = \{A_i\}_{i \in I}$ in $E$, the integral of the maximum expectation value over the pure states is equal to the upper envelope of $s$ evaluated at the barycenter $\phi$: $$\int_{\text{PureState}(E)} \max_{i \in I} \omega(A_i) \, d\mu(\omega) = \hat{s}(\phi)$$ where $\hat{s}$ denotes the upper envelope of the family of observables.

Given an Archimedean order-unit space $E$ and its state space $\mathcal{S}(E)$, `maxApplyDiffs` is the set of continuous real-valued functions on $\mathcal{S}(E)$ that can be expressed as the difference between two maximum expectation value functions. Specifically, a function $g$ belongs to this set if there exist finite non-empty index sets $I$ and $J$, and families of observables $\{A_i\}_{i \in I}$ and $\{B_j\}_{j \in J}$ in $E$, such that for every state $\omega \in \mathcal{S}(E)$: $$g(\omega) = \max_{i \in I} \omega(A_i) - \max_{j \in J} \omega(B_j)$$

Let $E$ be an Archimedean order-unit space and $\mathcal{S}(E)$ be its state space. Let $\mathcal{D} \subset C(\mathcal{S}(E), \mathbb{R})$ be the set of continuous real-valued functions (known as `maxApplyDiffs`) that can be expressed as the difference between two maximum expectation value functions. That is, $f \in \mathcal{D}$ if there exist finite non-empty index sets $I, J$ and families of observables $\{A_i\}_{i \in I}, \{B_j\}_{j \in J}$ in $E$ such that for every state $\omega \in \mathcal{S}(E)$: $$f(\omega) = \max_{i \in I} \omega(A_i) - \max_{j \in J} \omega(B_j)$$ For any two functions $g, h \in \mathcal{D}$, their pointwise supremum $g \sqcup h$, defined by $(g \sqcup h)(\omega) = \max(g(\omega), h(\omega))$, is also an element of $\mathcal{D}$.

Let $E$ be an Archimedean order-unit space and let $\mathcal{S}(E)$ be its state space. Let $C(\mathcal{S}(E), \mathbb{R})$ be the space of continuous real-valued functions on the state space. The set $\text{maxApplyDiffs}$ consists of functions $g \in C(\mathcal{S}(E), \mathbb{R})$ that can be expressed as the difference of two maximum expectation value functions: $$g(\omega) = \max_{i \in I} \omega(A_i) - \max_{j \in J} \omega(B_j)$$ where $I$ and $J$ are finite non-empty index sets and $\{A_i\}_{i \in I}, \{B_j\}_{j \in J}$ are families of observables in $E$. This theorem states that if a function $g$ belongs to $\text{maxApplyDiffs}$, then its negation $-g$ also belongs to $\text{maxApplyDiffs}$.

Let $E$ be an Archimedean order-unit space and $\mathcal{S}(E)$ be its state space. Let $\mathcal{D}$ be the set of continuous real-valued functions on $\mathcal{S}(E)$ that can be expressed as the difference between two maximum expectation value functions; that is, a function $f$ is in $\mathcal{D}$ if there exist finite non-empty index sets $I$ and $J$ and families of observables $\{A_i\}_{i \in I}$ and $\{B_j\}_{j \in J}$ in $E$ such that for every state $\omega \in \mathcal{S}(E)$, $$f(\omega) = \max_{i \in I} \omega(A_i) - \max_{j \in J} \omega(B_j).$$ For any two functions $g, h \in \mathcal{D}$, their pointwise infimum $g \sqcap h$, defined by $(g \sqcap h)(\omega) = \min(g(\omega), h(\omega))$, is also an element of $\mathcal{D}$.

Let $E$ be an Archimedean order-unit space and $\mathcal{S}(E)$ be its state space. Let $\mathcal{D}$ be the set of continuous functions $g: \mathcal{S}(E) \to \mathbb{R}$ that can be expressed as the difference of two maximum expectation values, i.e., $$g(\omega) = \max_{i \in I} \omega(A_i) - \max_{j \in J} \omega(B_j)$$ for some finite non-empty index sets $I, J$ and observables $A_i, B_j \in E$. Then the set $\mathcal{D}$ strongly separates the points of $\mathcal{S}(E)$. This means that for any two distinct states $\omega_1, \omega_2 \in \mathcal{S}(E)$, there exists a function $g \in \mathcal{D}$ such that $g(\omega_1) \neq g(\omega_2)$, and for every state $\omega \in \mathcal{S}(E)$, there exists a function $g \in \mathcal{D}$ such that $g(\omega) \neq 0$.

Let $E$ be an Archimedean order-unit space and let $\mathcal{S}(E)$ be its state space. Let $C(\mathcal{S}(E), \mathbb{R})$ be the space of continuous real-valued functions on $\mathcal{S}(E)$ equipped with the topology of uniform convergence. Let $\mathcal{D}$ be the set of functions $g: \mathcal{S}(E) \to \mathbb{R}$ of the form $$g(\omega) = \max_{i \in I} \omega(A_i) - \max_{j \in J} \omega(B_j)$$ where $I$ and $J$ are finite non-empty index sets and $A_i, B_j \in E$ are observables. Then the topological closure of $\mathcal{D}$ is the entire space $C(\mathcal{S}(E), \mathbb{R})$. That is, the differences of maximum expectation values are dense among continuous functions on the state space.

Let $E$ be an Archimedean order-unit space with state space $\text{stateSpace}(E)$. Let $\mu$ and $\nu$ be finite measures on the set of pure states $\text{PureState}(E) \subseteq \text{stateSpace}(E)$. If $\mu$ and $\nu$ represent the same positive linear functional on $E$ (i.e., $\text{toPositive } \mu = \text{toPositive } \nu$, meaning $\int \omega(A) \, d\mu(\omega) = \int \omega(A) \, d\nu(\omega)$ for all $A \in E$), then for any continuous function $g: \text{stateSpace}(E) \to \mathbb{R}$, the integral of $g$ with respect to the pushforward of $\mu$ to the state space equals the integral of $g$ with respect to the pushforward of $\nu$: $$\int_{\text{stateSpace}(E)} g(\omega) \, d(\iota_* \mu)(\omega) = \int_{\text{stateSpace}(E)} g(\omega) \, d(\iota_* \nu)(\omega)$$ where $\iota: \text{PureState}(E) \hookrightarrow \text{stateSpace}(E)$ is the inclusion map.

Let $E$ be an Archimedean order-unit space. Suppose $E$ satisfies the ensemble refinement property, meaning that for any two finite families of positive linear functionals $\{\alpha_i\}_{i \in \mathcal{I}}$ and $\{\beta_j\}_{j \in \mathcal{J}}$ on $E$ such that $\sum_{i \in \mathcal{I}} \alpha_i = \sum_{j \in \mathcal{J}} \beta_j$, there exists a common refinement $\{\gamma_{ij}\}_{i \in \mathcal{I}, j \in \mathcal{J}}$. If a state $\omega \in \mathcal{S}(E)$ has a pure decomposition (i.e., there exists a regular probability measure $\mu$ on the set of pure states of $E$ such that $\omega(f) = \int_{\text{PureState } E} \phi(f) \, d\mu(\phi)$ for all $f \in E$), then this decomposition is unique.