PhyslibAlpha.ProbabilisticTheory.Classical.Nuclear

Let $E$ be an order-unit space and let $\text{Bidual}(E)$ be its bidual space. For any sequence $(x_n)_{n \in \mathbb{N}}$ in $\text{Bidual}(E)$ that is monotonically increasing ($x_n \le x_{n+1}$ for all $n$) and bounded above by the order unit $1$ ($x_n \le 1$ for all $n$), the least upper bound $\sup_{n \in \mathbb{N}} x_n$ exists in $\text{Bidual}(E)$.

Let $E$ be an order-unit space and $E^{**}$ denote its bidual space with order unit $1$. Suppose $c \in E^{**}$ is a non-zero positive element ($c \ge 0, c \neq 0$) that is disjoint from its complement in the sense that $c \wedge (1 - c) = 0$. For any $\theta > 0$, there exists a state $\phi$ on $E$ (a positive linear functional such that $\phi(1) = 1$) such that the evaluation of $c$ at $\phi$ satisfies $c(\phi) \ge 1 - \theta$.

Let $E$ be an order-unit space and $\text{Bidual}(E)$ be its bidual space. Suppose $s = \{s_i\}_{i \in I}$ is a partition of the unit $1$ in $\text{Bidual}(E)$. For any $\theta > 0$, there exists a family of states $\{\phi_i\}_{i \in I}$ (where each $\phi_i$ is a positive linear functional on $E$ satisfying $\phi_i(1) = 1$) such that for every index $i$ where $s_i \neq 0$, the action of the bidual element $s_i$ on the state $\phi_i$ satisfies $s_i(\phi_i) \geq 1 - \theta$.

Let $E$ be a physical system and $E^{**}$ denote its bidual. For a collection of observables $s_i \in E^{**}$ and real coefficients $c_i \in \mathbb{R}$, let $\sum_i c_i s_i$ be a linear combination (referred to as a step function in this context). For any state $\phi$ of the system, the evaluation of this sum at $\phi$ is equal to the linear combination of the individual evaluations: $$\left( \sum_i c_i s_i \right)(\phi) = \sum_i c_i (s_i(\phi))$$ where the pairing $(\cdot)(\phi)$ represents the action of the bidual element on the state.

For any finite family of elements $\{s_i\}_{i \in I}$ in the bidual space $E^{**}$ and any positive linear functional $\phi$ on $E$, the evaluation of the sum $\sum_{i \in I} s_i$ at $\phi$ is equal to the sum of the individual evaluations: $$\left( \sum_{i \in I} s_i \right)(\phi) = \sum_{i \in I} s_i(\phi)$$

Let $E$ be an order-unit space and $E^{**}$ its bidual. Let $\{s_i\}_{i \in I}$ be a partition of the unit $1 \in E^{**}$ and let $x \in E^{**}$ be an observable in the bidual. Suppose $x$ is bounded between two step functions such that $\sum_{j \in I} a_j s_j \le x \le \sum_{j \in I} b_j s_j$ for some real coefficients $a_j, b_j$ that are all bounded in absolute value by $A$ (i.e., $|a_l| \le A$ and $|b_l| \le A$ for all $l \in I$). If $\phi$ is a normalized state ($\phi(1) = 1$) that detects the $i$-th piece of the partition with error $\theta$ (meaning $s_i(\phi) \ge 1 - \theta$), then the evaluation of $x$ at the state $\phi$ is bounded by: $$a_i - 2A\theta \le x(\phi) \le b_i + 2A\theta$$

For any finite collection of real numbers $a_{i,k}$ (where $i \in \{0, \dots, m-1\}$ and $k \in \{0, \dots, n-1\}$) and any real constant $c$, there exists a non-negative real number $A$ such that for all indices $i$ and $k$, the absolute values of both $a_{i,k}$ and the shifted values $a_{i,k} + c$ are bounded by $A$, i.e., $|a_{i,k}| \le A$ and $|a_{i,k} + c| \le A$.

For any positive real number $\eta > 0$ and any non-negative real number $A \ge 0$, there exists a positive real number $\theta > 0$ such that $2 A \theta \le \frac{\eta}{4}$. This $\theta$ serves as a detection error parameter in approximation arguments.

Let $E$ be a classical order-unit space and let $\text{Bidual}(E)$ be its bidual space. Given a finite collection of elements $x_1, \dots, x_n \in \text{Bidual}(E)$ and a positive real number $\eta > 0$, there exists a finite number $m$, a set of non-negative elements $p_1, \dots, p_m \in \text{Bidual}(E)$ forming a partition of the unit (i.e., $\sum_{i=1}^m p_i = 1$), and a set of states $\varphi_1, \dots, \varphi_m$ on $E$ such that for each $k \in \{1, \dots, n\}$, the following inequality holds: $$-\eta \cdot 1 \le x_k - \sum_{i=1}^m \langle x_k, \varphi_i \rangle p_i \le \eta \cdot 1$$ where $\langle x_k, \varphi_i \rangle$ denotes the natural pairing between the bidual element $x_k$ and the state $\varphi_i$, and $1$ is the order unit of the bidual space.

Given ordered real vector spaces $E$ and $F$, and a linear functional $\Lambda: E \otimes F \to \mathbb{R}$ that is positive on the minimal tensor cone, for any fixed non-negative element $y \in F$ ($y \ge 0$), the `slice` is the positive linear functional on $E$ mapping $x$ to $\Lambda(x \otimes y)$.

Let $E$ and $F$ be ordered real vector spaces. Let $\Lambda : E \otimes_{\mathbb{R}} F \to \mathbb{R}$ be a linear functional that is positive on the minimal tensor cone. For any non-negative element $y \in F$ (where $y \ge 0$) and any element $x \in E$, the value of the positive functional $\text{slice}(\Lambda, y)$ applied to $x$ is equal to $\Lambda(x \otimes y)$.

Let $E$ and $F$ be ordered real vector spaces and let $\Lambda: E \otimes F \to \mathbb{R}$ be a linear functional that is non-negative on the minimal tensor cone. For any non-negative element $y \in F$, let the "slice" be the linear functional on $E$ defined by $x \mapsto \Lambda(x \otimes y)$. Then for any non-negative elements $y, y' \in F$, the slice associated with the sum $y + y'$ is equal to the sum of the slice associated with $y$ and the slice associated with $y'$.

Let $E$ and $F$ be ordered real vector spaces and let $\Lambda: E \otimes_{\mathbb{R}} F \to \mathbb{R}$ be a linear functional that is non-negative on the minimal tensor cone. For any $y \in F$ with $y \ge 0$, let $\text{slice}(y)$ be the positive linear functional on $E$ defined by $x \mapsto \Lambda(x \otimes y)$. For any scalar $c \in \mathbb{R}$ such that $c \ge 0$ and any $y \in F$ such that $y \ge 0$, it holds that $$\text{slice}(c \cdot y) = c \cdot \text{slice}(y).$$

Given a linear functional $\Lambda: E \otimes F \to \mathbb{R}$ that is positive on the minimal tensor cone, the function `Nuclear.pairing` maps a non-negative element $p$ of the bidual $E^{**}$ to a positive linear functional on $F$. For any $y \ge 0$ in $F$, the value of the resulting functional is $p(\phi_y)$, where $\phi_y: E \to \mathbb{R}$ is the positive linear functional defined by $\phi_y(x) = \Lambda(x \otimes y)$.

Let $E$ and $F$ be ordered real vector spaces and let $\Lambda: E \otimes_{\mathbb{R}} F \to \mathbb{R}$ be a linear functional that is non-negative on the minimal tensor cone. For any non-negative element $p$ in the bidual $E^{**}$ and any non-negative element $y \in F$, the value of the positive linear functional $\text{pairing}(p)$ at $y$ is equal to the result of evaluating $p$ on the functional $\text{slice}(y)$. That is, $$\text{pairing}(p)(y) = p(\text{slice}(y)),$$ where $\text{slice}(y)$ is the linear functional on $E$ defined by $x \mapsto \Lambda(x \otimes y)$.

Let $E$ and $F$ be ordered real vector spaces (specifically order-unit spaces), and let $\Lambda: E \otimes F \to \mathbb{R}$ be a linear functional that is positive on the minimal tensor cone. Suppose an element $e \in E$ has its canonical image in the bidual $E^{**}$ decomposed as $\text{ev}(e) = \sum_{i \in I} c_i p_i + d$, where $I$ is a finite index set, $p_i \in E^{**}$ are non-negative elements ($p_i \ge 0$), $c_i \in \mathbb{R}$ are coefficients, and $d \in E^{**}$ is a remainder. For any $f \in F$ and any $N \in \mathbb{R}$ such that $N \ge 0$ and $f + N \cdot \mathbf{1} \ge 0$ (where $\mathbf{1}$ is the order unit), the value of $\Lambda$ on the elementary tensor $e \otimes f$ is given by: $$\Lambda(e \otimes f) = \sum_{i \in I} c_i \cdot \text{pairing}(\Lambda, p_i, f) + d(\text{slice}(\Lambda, f + N \cdot \mathbf{1})) - d(\text{slice}(\Lambda, N \cdot \mathbf{1}))$$ where $\text{pairing}(\Lambda, p, f)$ is the evaluation of the bidual element $p$ on the positive linear functional $x \mapsto \Lambda(x \otimes f)$, and $\text{slice}(\Lambda, y)$ is the linear functional on $E$ defined by $x \mapsto \Lambda(x \otimes y)$ for $y \ge 0$.

Let $E$ and $F$ be ordered real vector spaces. If an element $z \in E \otimes_{\mathbb{R}} F$ belongs to the maximal tensor cone, then for any positive linear functional $\phi : E \to \mathbb{R}$, the left slice of $z$ by $\phi$ is non-negative in $F$. That is, $0 \le \text{lslice } \phi(z)$.

Let $E$ and $F$ be ordered real vector spaces, and let $\Lambda: E \otimes_{\mathbb{R}} F \to \mathbb{R}$ be a linear functional that is non-negative on the minimal tensor cone. Let $z = \sum_{k=1}^n e_k \otimes f_k$ be an element of the maximal tensor cone of $E \otimes_{\mathbb{R}} F$. Given any finite collection of non-negative elements $\{p_i\}_{i=1}^m$ in the bidual space $E^{**}$ and any finite collection of positive linear functionals $\{\phi_i\}_{i=1}^m$ on $E$, the "measured-and-prepared" sum is non-negative: $$\sum_{k=1}^n \sum_{i=1}^m \phi_i(e_k) \cdot \text{pairing}(\Lambda, p_i, f_k) \ge 0$$ where $\text{pairing}(\Lambda, p_i, f_k)$ is the value at $f_k$ of the functional on $F$ induced by $p_i$ and $\Lambda$ (specifically, the evaluation of $p_i$ on the functional $x \mapsto \Lambda(x \otimes f_k)$), and $\phi_i(e_k)$ is the evaluation of $e_k$ under the functional $\phi_i$.

Let $E$ be an order-unit space and $\text{Bidual}(E)$ its bidual space. Let $1$ denote the order unit in $\text{Bidual}(E)$. For any element $d \in \text{Bidual}(E)$ and real number $\eta$ such that $-\eta \cdot 1 \le d \le \eta \cdot 1$, and for any two positive linear functionals $\psi_1, \psi_2$ on $E$, the difference in the evaluations of $d$ on these functionals is bounded below by: $$d(\psi_1) - d(\psi_2) \ge -\eta (\psi_1(1) + \psi_2(1))$$ where $\psi(1)$ denotes the weight (evaluation) of the functional $\psi$ on the order unit $1 \in E$.

Let $E$ be a classical physical system and $F$ be an ordered real vector space. Let $\Lambda$ be a linear functional on the tensor product $E \otimes_{\mathbb{R}} F$ that is nonnegative on the minimal tensor cone (the cone generated by elementary tensors $x \otimes y$ where $x \in E$ and $y \in F$ are nonnegative). If $z$ is an element of the maximal tensor cone of $E \otimes_{\mathbb{R}} F$, then it holds that $\Lambda(z) \ge 0$.

If the positive functionals on a physical system $E$ form a lattice (i.e., $E$ is a classical system), then $E$ is nuclear. This means that for any Archimedean system $F$, the composition of $E$ and $F$ is unique in the sense that every composite observable in the maximal tensor cone lies in the closure of the minimal tensor cone.

An Archimedean order-unit space $E$ is classical—meaning its state space is a Choquet simplex and its positive functionals form a lattice—if and only if it is nuclear, meaning that for any other Archimedean system, the minimal and maximal tensor cones coincide (it composes uniquely).