PhyslibAlpha.ProbabilisticTheory.Composite.TensorCone

Given two physical systems with observable spaces $E$ and $F$, the space of observables for the composite system is defined as the tensor product $E \otimes_{\mathbb{R}} F$.

For two physical systems with observable spaces $E$ and $F$, this function maps an observable $x \in E$ and an observable $y \in F$ to their tensor product $x \otimes y$ in the composite observable space $E \otimes_{\mathbb{R}} F$.

For a positive linear functional $\psi : F \to \mathbb{R}$, the right slice map is the linear map $E \otimes F \to E$ obtained by applying $\psi$ to the second factor of the tensor product and identifying $E \otimes \mathbb{R}$ with $E$. For an elementary tensor $x \otimes y \in E \otimes F$, the map is defined by $\text{rslice}(\psi)(x \otimes y) = \psi(y) \cdot x$.

Given a positive linear functional $\phi: E \to \mathbb{R}$, the left slice map $\text{lslice } \phi$ is the linear map from the tensor product $E \otimes F$ to $F$ obtained by applying $\phi$ to the first factor. For any $x \in E$ and $y \in F$, the map is defined on basic tensors by $(\text{lslice } \phi)(x \otimes y) = \phi(x) \cdot y$.

Let $E$ and $F$ be vector spaces over $\mathbb{R}$. For a positive linear functional $\psi : F \to \mathbb{R}$ and elements $x \in E$ and $y \in F$, the right slice map $\text{rslice}(\psi) : E \otimes F \to E$ applied to the elementary tensor $x \otimes y$ satisfies: $$\text{rslice}(\psi)(x \otimes y) = \psi(y) \cdot x$$

Let $E$ and $F$ be vector spaces over $\mathbb{R}$. For any positive linear functional $\phi: E \to \mathbb{R}$ and any elements $x \in E$ and $y \in F$, the left slice map $\text{lslice } \phi$ applied to the elementary tensor $x \otimes y$ is equal to the scalar $\phi(x)$ multiplied by the vector $y$: $$(\text{lslice } \phi)(x \otimes y) = \phi(x) \cdot y$$

Given two positive linear functionals $\phi: E \to \mathbb{R}$ and $\psi: F \to \mathbb{R}$, the product functional $\phi \otimes \psi$ is the linear map from the tensor product $E \otimes F$ to $\mathbb{R}$ defined by the composition $\phi \circ \text{rslice}(\psi)$. For any elementary tensor $x \otimes y \in E \otimes F$, the functional satisfies: $$(\phi \otimes \psi)(x \otimes y) = \phi(x) \cdot \psi(y)$$ This functional represents the joint preparation or product state in a composite system.

Let $E$ and $F$ be vector spaces over $\mathbb{R}$. For any positive linear functionals $\phi: E \to \mathbb{R}$ and $\psi: F \to \mathbb{R}$, and for any elements $x \in E$ and $y \in F$, the product functional $\phi \otimes \psi$ applied to the elementary tensor $x \otimes y$ is equal to the product of the values $\phi(x)$ and $\psi(y)$: $$(\phi \otimes \psi)(x \otimes y) = \phi(x) \psi(y)$$

For positive linear functionals $\phi: E \to \mathbb{R}$ and $\psi: F \to \mathbb{R}$, and for any element $z \in E \otimes F$ of the tensor product, the product functional $\phi \otimes \psi$ evaluated at $z$ satisfies: $$(\phi \otimes \psi)(z) = \phi(\text{rslice}(\psi)(z))$$ where $\text{rslice}(\psi) : E \otimes F \to E$ is the right slice map obtained by applying $\psi$ to the second factor of the tensor product.

Let $E$ and $F$ be vector spaces over $\mathbb{R}$. For any positive linear functionals $\phi: E \to \mathbb{R}$ and $\psi: F \to \mathbb{R}$, and for any element $z$ in the tensor product $E \otimes F$, the product functional $\phi \otimes \psi$ applied to $z$ is equal to the functional $\psi$ applied to the left slice of $z$ by $\phi$. That is, $$(\phi \otimes \psi)(z) = \psi(\text{lslice } \phi(z))$$

Given two ordered real vector spaces $E$ and $F$, the minimal tensor cone is a pointed cone in the tensor product $E \otimes_{\mathbb{R}} F$ consisting of all finite sums of elementary tensors $x \otimes y$ where $x \in E$ and $y \in F$ are non-negative elements (i.e., $x \ge 0$ and $y \ge 0$).

Let $E$ and $F$ be ordered real vector spaces representing physical systems. The maximal tensor cone, denoted as $\text{maxTensorCone}(E, F)$, is the pointed cone in the algebraic tensor product $E \otimes_{\mathbb{R}} F$ consisting of all composite observables $z$ that are non-negative under every product of positive linear functionals. That is, $z \in \text{maxTensorCone}(E, F)$ if and only if for all positive linear functionals $\phi: E \to \mathbb{R}$ and $\psi: F \to \mathbb{R}$, the product functional satisfies $(\phi \otimes \psi)(z) \ge 0$.

In the context of composite systems represented by ordered real vector spaces $E$ and $F$, let $x \in E$ and $y \in F$ be non-negative elements (i.e., $x \ge 0$ and $y \ge 0$). Then their elementary tensor product $x \otimes y$ is an element of the minimal tensor cone of the algebraic tensor product $E \otimes_{\mathbb{R}} F$.

Let $E$ and $F$ be ordered real vector spaces and $E \otimes_{\mathbb{R}} F$ be their algebraic tensor product. Let the minimal tensor cone, denoted as $\text{minTensorCone}(E, F)$, be the cone in $E \otimes_{\mathbb{R}} F$ generated by the elementary tensors of non-negative elements. For any non-negative real scalar $c \ge 0$ and any element $w \in \text{minTensorCone}(E, F)$, the scalar product $c \cdot w$ is also an element of $\text{minTensorCone}(E, F)$.

Let $E$ and $F$ be ordered real vector spaces representing physical systems, and let $E \otimes F$ be their composite system's space of observables. Let $p$ be a property of observables in $E \otimes F$ and $z$ be an observable in the minimal tensor cone. If the following conditions hold: - $p(x \otimes y)$ is true for all non-negative observables $x \in E$ and $y \in F$, - $p(0)$ is true, - $p(z + w)$ is true whenever $p(z)$ and $p(w)$ are true, - $p(c \cdot z)$ is true for all $c \ge 0$ whenever $p(z)$ is true, then $p(z)$ is true for all $z$ in the minimal tensor cone.

Let $E$ and $F$ be ordered real vector spaces representing physical systems. Given two positive linear functionals $\phi: E \to \mathbb{R}$ and $\psi: F \to \mathbb{R}$, let $\phi \otimes \psi$ denote their product functional on the composite system $E \otimes_{\mathbb{R}} F$. For any element $z \in E \otimes_{\mathbb{R}} F$, the value of the product functional $(\phi \otimes \psi)(z)$ is equal to the value obtained by applying the tensor product of the linear maps $\phi$ and $\psi$ (viewed as an element of the dual tensor space $E^* \otimes_{\mathbb{R}} F^*$) to $z$ via the canonical distribution map. Mathematically, this identity is: $$(\phi \otimes \psi)(z) = \text{dualDistrib}(\phi \otimes \psi)(z)$$ where $\text{dualDistrib}$ is the natural map from the tensor product of dual spaces to the dual of the tensor product, $E^* \otimes F^* \to (E \otimes F)^*$.

Let $E$ and $F$ be ordered real vector spaces representing physical systems. For any composite observable $z \in E \otimes_{\mathbb{R}} F$, $z$ belongs to the maximal tensor cone $\text{maxTensorCone}(E, F)$ if and only if for all positive linear functionals $\phi: E \to \mathbb{R}$ and $\psi: F \to \mathbb{R}$, the product functional $(\phi \otimes \psi)$ evaluates to a non-negative result on $z$: $$(\phi \otimes \psi)(z) \ge 0$$

Let $E$ and $F$ be ordered real vector spaces representing physical systems. The minimal tensor cone $\text{minTensorCone}(E, F)$ is a subset of the maximal tensor cone $\text{maxTensorCone}(E, F)$. Mathematically, $$\text{minTensorCone}(E, F) \subseteq \text{maxTensorCone}(E, F)$$ This means that every composite observable in the minimal cone (any finite sum of elementary tensors $x \otimes y$ where $x \ge 0$ and $y \ge 0$) is also contained in the maximal cone (the set of observables $z$ such that $(\phi \otimes \psi)(z) \ge 0$ for all positive linear functionals $\phi: E \to \mathbb{R}$ and $\psi: F \to \mathbb{R}$).

Let $E$ and $E'$ be ordered vector spaces over $\mathbb{R}$. For any positive linear map $\phi : E \to E'$, the image of the positive cone of $E$ under $\phi$ is contained in the positive cone of $E'$. Mathematically, if $\text{PosCone}(E) = \{x \in E \mid 0 \le x\}$, then $\phi(\text{PosCone}(E)) \subseteq \text{PosCone}(E')$.

Let $E, E', F,$ and $F'$ be ordered real vector spaces. If $\phi : E \to E'$ and $\psi : F \to F'$ are positive linear maps, then the image of the minimal tensor cone of $E \otimes F$ under the tensor product map $\phi \otimes \psi$ is contained in the minimal tensor cone of $E' \otimes F'$.

Let $E, E', F,$ and $F'$ be ordered real vector spaces. Given positive linear maps $\phi : E \to E'$ and $\psi : F \to F'$, the image of the maximal tensor cone of $E \otimes_{\mathbb{R}} F$ under the tensor product map $\phi \otimes \psi : E \otimes_{\mathbb{R}} F \to E' \otimes_{\mathbb{R}} F'$ is contained in the maximal tensor cone of $E' \otimes_{\mathbb{R}} F'$. That is, $$(\phi \otimes \psi)(\text{maxTensorCone}(E, F)) \subseteq \text{maxTensorCone}(E', F')$$ where the maximal tensor cone of $A \otimes B$ consists of those tensors $u$ such that $(\omega_A \otimes \omega_B)(u) \ge 0$ for all positive linear functionals $\omega_A: A \to \mathbb{R}$ and $\omega_B: B \to \mathbb{R}$.

Let $E, E', F,$ and $F'$ be ordered real vector spaces. Let $\phi: E \to E'$ and $\psi: F \to F'$ be positive linear maps. If an element $z$ of the algebraic tensor product $E \otimes_{\mathbb{R}} F$ lies in the minimal tensor cone of $E \otimes_{\mathbb{R}} F$, then its image under the tensor product map $\phi \otimes \psi$ lies in the minimal tensor cone of $E' \otimes_{\mathbb{R}} F'$.

Let $E, E', F,$ and $F'$ be ordered real vector spaces representing systems of observables. Let $\phi: E \to E'$ and $\psi: F \to F'$ be positive linear maps. If a composite observable $z$ in the tensor product $E \otimes_{\mathbb{R}} F$ belongs to the maximal tensor cone of $E$ and $F$, then its image under the tensor product map, $(\phi \otimes \psi)(z)$, belongs to the maximal tensor cone of $E'$ and $F'$.

A pointed cone $C \subseteq E \otimes_{\mathbb{R}} F$ of composite observables is **Archimedean** if, for every $z \in E \otimes_{\mathbb{R}} F$, the condition that $z + \epsilon (1_E \otimes 1_F) \in C$ for all $\epsilon > 0$ implies that $z \in C$, where $1_E$ and $1_F$ are the identity elements of the systems $E$ and $F$ respectively.

The structure `CompositeCone E F` is treated as a set of elements within the tensor product $E \otimes_{\mathbb{R}} F$. This enables the use of set-theoretic notation for composite cones, such that for any $C \in \text{CompositeCone } E F$ and any composite observable $z \in E \otimes_{\mathbb{R}} F$, the statement $z \in C$ is well-defined and corresponds to $z$ belonging to the underlying pointed cone of $C$.

Given a composite cone $C$ of the systems $E$ and $F$, a composite observable $z \in E \otimes_{\mathbb{R}} F$ belongs to the underlying pointed cone of $C$ if and only if $z \in C$. Here, the notation $z \in C$ refers to the membership of $z$ in the set of nonnegative observables defined by the composite cone structure.

Let $E$ and $F$ be systems of observables, and let $E \otimes_{\mathbb{R}} F$ be their tensor product space. For any two composite cones $C, D \in \text{CompositeCone } E F$, if for every composite observable $z \in E \otimes_{\mathbb{R}} F$ it holds that $z \in C$ if and only if $z \in D$, then the composite cones are equal ($C = D$).

Let $E$ and $F$ be physical systems with observables. For any composite cone $C$ associated with the composite system $E \otimes F$, if $x \in E$ and $y \in F$ are nonnegative observables ($x \ge 0$ and $y \ge 0$), then their tensor product $x \otimes y$ is an element of $C$.

Let $E$ and $F$ be systems and let $C$ be a composite cone on the tensor product $E \otimes_{\mathbb{R}} F$. For any positive linear functionals $\phi: E \to \mathbb{R}$ and $\psi: F \to \mathbb{R}$, and for any composite observable $z \in C$, the product functional $\phi \otimes \psi$ is non-negative on $z$: $$(\phi \otimes \psi)(z) \ge 0$$

Given a composite cone $C$ of systems $E$ and $F$, the set of elements in $C$ (viewed as a pointed cone) forms a module over the semiring of nonnegative real numbers $\mathbb{R}_{\ge 0}$. This structure inherits the addition and scalar multiplication from the underlying tensor product space $E \otimes_{\mathbb{R}} F$.

For two systems with ordered real vector spaces of observables $E$ and $F$, the minimal composite cone is the composite structure on the tensor product $E \otimes_{\mathbb{R}} F$ whose positive cone is the minimal tensor cone. This cone is generated by the set of elementary tensors $x \otimes y$ where $x$ and $y$ are nonnegative observables in $E$ and $F$ respectively.

For two ordered real vector spaces $E$ and $F$ representing physical systems, the maximal composite cone is the composite structure on the tensor product $E \otimes_{\mathbb{R}} F$ whose set of nonnegative elements is defined by the maximal tensor cone. This cone consists of all tensors $u \in E \otimes_{\mathbb{R}} F$ such that for every pair of positive linear functionals $\phi: E \to \mathbb{R}$ and $\psi: F \to \mathbb{R}$, the product functional $\phi \otimes \psi$ yields a non-negative result, i.e., $(\phi \otimes \psi)(u) \ge 0$.

A composite cone $C \subseteq E \otimes F$ is **Archimedean** if for every composite observable $z \in E \otimes F$, the condition that $z + \epsilon (1_E \otimes 1_F) \in C$ for all $\epsilon > 0$ implies that $z \in C$. Here, $1_E$ and $1_F$ denote the identity elements of the systems $E$ and $F$ respectively.

Given a composite cone $C$ of systems $E$ and $F$ and a fixed non-negative element $y_0 \in F$ (such that $y_0 \ge 0$), `tmulRight` is the $\mathbb{R}_{\ge 0}$-linear map from the positive cone of $E$ to the cone $C$ defined by $x \mapsto x \otimes y_0$. This map embeds non-negative observables of $E$ into the composite system by tensoring them on the right with the fixed observable $y_0$.

Let $E$ and $F$ be real vector spaces and $C \subseteq E \otimes_{\mathbb{R}} F$ be a composite cone. Given a non-negative element $y_0 \in F$ (i.e., $0 \le y_0$), let $\text{tmulRight}_{y_0} : \text{PosCone}(E) \to C$ be the $\mathbb{R}_{\ge 0}$-linear map defined by tensoring on the right with $y_0$. For any $x \in \text{PosCone}(E)$, the image of $x$ under this map is equal to the tensor product $x \otimes y_0$ in $E \otimes_{\mathbb{R}} F$.

Given a composite cone $C$ of systems $E$ and $F$ and a fixed non-negative element $x_0 \in E$ (such that $x_0 \ge 0$), `tmulLeft` is the $\mathbb{R}_{\ge 0}$-linear map from the positive cone of $F$ to the cone $C$ defined by $y \mapsto x_0 \otimes y$. This map embeds non-negative observables of $F$ into the composite system by tensoring them on the left with the fixed observable $x_0$.

Let $E$ and $F$ be systems and $C$ be a composite cone of $E$ and $F$. For any non-negative observable $x_0 \in E$ (where $x_0 \ge 0$) and any element $y$ in the positive cone $\text{PosCone}(F)$, the value of the $\mathbb{R}_{\ge 0}$-linear map $C.\text{tmulLeft}$ applied to $y$ is the tensor product of $x_0$ and $y$: $$(C.\text{tmulLeft } x_0)(y) = x_0 \otimes y$$ where $\otimes$ denotes the tensor product in $E \otimes_{\mathbb{R}} F$.

For any element $t$ in the tensor product $E \otimes_{\mathbb{R}} F$, there exist elements $t_p$ and $t_n$ in the minimal tensor cone of $E$ and $F$ such that $t = t_p - t_n$. In this context, the minimal tensor cone is the set of all finite sums of elementary tensors $x \otimes y$ where $x \in E$ and $y \in F$ are non-negative elements. Thus, every composite observable can be expressed as a difference of two sums of products of non-negative observables.

For any observables $x \in E$ and $y \in F$, there exists a real number $t$ such that the composite observable $x \otimes y + t(1 \otimes 1)$ belongs to the minimal tensor cone of the composite system $E \otimes F$. Specifically, for sufficiently large $a, b \in \mathbb{R}$ such that $a \cdot 1 \pm x \geq 0$ and $b \cdot 1 \pm y \geq 0$, the identity $x \otimes y + (ab) \cdot (1 \otimes 1) = \frac{1}{2}((a \cdot 1 + x) \otimes (b \cdot 1 + y) + (a \cdot 1 - x) \otimes (b \cdot 1 - y))$ shows that the expression is a sum of products of nonnegative observables.

Let $E$ and $F$ be ordered real vector spaces with identity observables $1_E$ and $1_F$. For any element $z$ in the algebraic tensor product $E \otimes_{\mathbb{R}} F$, there exists a real number $t$ such that the element $z + t(1_E \otimes 1_F)$ belongs to the minimal tensor cone of $E \otimes F$.

In the composite system of two ordered vector spaces $E$ and $F$ with units $1_E$ and $1_F$, the minimal tensor cone is dominated by the tensor product of the units $1_E \otimes 1_F$. That is, for any element $z \in E \otimes_{\mathbb{R}} F$, there exists a scalar $t \in \mathbb{R}$ such that $z + t(1_E \otimes 1_F)$ lies in the minimal tensor cone.

The Archimedean closure of the minimal tensor cone in the algebraic tensor product $E \otimes_{\mathbb{R}} F$ of two observable spaces $E$ and $F$. A composite observable $z \in E \otimes_{\mathbb{R}} F$ belongs to this set if, for every $\epsilon > 0$, the element $z + \epsilon (1_E \otimes 1_F)$ lies within the minimal tensor cone, where $1_E$ and $1_F$ are the identity elements of the respective systems.

The minimal tensor cone of the algebraic tensor product $E \otimes_{\mathbb{R}} F$ of two observable spaces $E$ and $F$ is a subset of its Archimedean closure.

In the algebraic tensor product $E \otimes_{\mathbb{R}} F$ of two observable spaces $E$ and $F$, the Archimedean closure of the minimal tensor cone is a subset of the maximal tensor cone. That is, $$\text{minTensorClosure}(E, F) \subseteq \text{maxTensorCone}(E, F).$$ This means that if an element $z \in E \otimes_{\mathbb{R}} F$ has the property that $z + \epsilon (1_E \otimes 1_F)$ lies in the minimal tensor cone for all $\epsilon > 0$, then it necessarily satisfies $(\phi \otimes \psi)(z) \geq 0$ for all pairs of positive linear functionals $\phi: E \to \mathbb{R}$ and $\psi: F \to \mathbb{R}$.

Let $E$ and $F$ be observable spaces and let $E \otimes_{\mathbb{R}} F$ be their algebraic tensor product. Let $C \subseteq E \otimes_{\mathbb{R}} F$ be a pointed cone that is Archimedean, meaning that for any $z \in E \otimes_{\mathbb{R}} F$, if $z + \epsilon (1_E \otimes 1_F) \in C$ for all $\epsilon > 0$, then $z \in C$. If $C$ contains the minimal tensor cone of $E \otimes_{\mathbb{R}} F$, then it also contains the Archimedean closure of the minimal tensor cone.

Let $E$ and $F$ be observable spaces and $E \otimes_{\mathbb{R}} F$ be their algebraic tensor product. If a composite cone $C \subseteq E \otimes_{\mathbb{R}} F$ is Archimedean, then it contains the Archimedean closure of the minimal tensor cone. That is, $$\text{minTensorClosure}(E, F) \subseteq C.$$ A composite cone $C$ is said to be Archimedean if for every composite observable $z \in E \otimes_{\mathbb{R}} F$, the condition that $z + \epsilon (1_E \otimes 1_F) \in C$ for all $\epsilon > 0$ implies that $z \in C$, where $1_E$ and $1_F$ are the identity elements of the respective systems.

A system $E$ is said to be **nuclear** if for every Archimedean system $F$, the maximal tensor cone is contained within the Archimedean closure of the minimal tensor cone. That is, for any Archimedean system $F$, we have the inclusion: $$\text{maxTensorCone}(E, F) \subseteq \text{minTensorClosure}(E, F).$$ This condition ensures that the choice of a composite cone between the minimal and maximal cones is unique up to Archimedean closure.

Let $E$ be an Archimedean ordered real vector space and $F$ be an ordered real vector space. An element $z$ in the tensor product $E \otimes_{\mathbb{R}} F$ belongs to the maximal tensor cone if and only if for every positive linear functional $\psi : F \to \mathbb{R}$, the right slice $\text{rslice}(\psi)(z)$ is a non-negative element of $E$. Here, the right slice $\text{rslice}(\psi)$ is the linear map $E \otimes F \to E$ defined on elementary tensors by $\text{rslice}(\psi)(x \otimes y) = \psi(y) \cdot x$.