PhyslibAlpha.ProbabilisticTheory.CStarAlgebra.MatrixComposite

Given a matrix $h \in \text{Mat}_{n \times n}(\mathbb{C})$ and an element $a$ of a $C^*$-algebra $A$, the function constructs a matrix in $\text{Mat}_{n \times n}(A)$ where each entry at index $(i, j)$ is the scalar product $h_{ij} \cdot a$. This represents the composite observable formed by the tensor product $a \otimes h$ of the observable $a$ with the scalar matrix $h$.

Let $A$ be a $C^*$-algebra and $h \in \text{Mat}_{n \times n}(\mathbb{C})$ be a complex matrix. For any element $a \in A$ and indices $i, j$, the $(i, j)$-th entry of the matrix $a \otimes h$ (denoted by `kron h a`) is equal to $h_{ij} \cdot a$, where $h_{ij}$ is the entry of $h$ at row $i$ and column $j$, and $\cdot$ denotes the scalar multiplication of an element of $A$ by a complex number.

Let $h, h' \in \text{Mat}_{n \times n}(\mathbb{C})$ be complex matrices and $a \in A$ be an element of a $C^*$-algebra $A$. The Kronecker product (or composite observable) $\text{kron}(h + h', a)$ is equal to the sum of the Kronecker products $\text{kron}(h, a)$ and $\text{kron}(h', a)$, where $\text{kron}(h, a)$ denotes the matrix in $\text{Mat}_{n \times n}(A)$ with entries $h_{ij} \cdot a$.

Let $A$ be a $C^*$-algebra and $h \in \text{Mat}_{n \times n}(\mathbb{C})$ be a complex $n \times n$ matrix. For any elements $a, a' \in A$, the Kronecker product of the observable and the matrix satisfies the distributive law: $$(a + a') \otimes h = (a \otimes h) + (a' \otimes h)$$ where $a \otimes h$ denotes the matrix in $\text{Mat}_{n \times n}(A)$ whose entries are given by $(a \otimes h)_{ij} = h_{ij} \cdot a$.

For any real number $r$, any $n \times n$ complex matrix $h \in \text{Mat}_{n \times n}(\mathbb{C})$, and any element $a$ of a $C^*$-algebra $A$, the Kronecker product of the scaled matrix $r \cdot h$ and $a$ is equal to the scalar multiplication of $r$ with the Kronecker product of $h$ and $a$: $$\text{kron}(r \cdot h, a) = r \cdot \text{kron}(h, a)$$

For any real scalar $r \in \mathbb{R}$, any $n \times n$ complex matrix $h \in \text{Mat}_{n \times n}(\mathbb{C})$, and any element $a$ of a $C^*$-algebra $A$, the Kronecker product of $h$ with the scaled element $r \cdot a$ is equal to the scaling of the Kronecker product of $h$ and $a$ by $r$. That is, $\text{kron}(h, r \cdot a) = r \cdot \text{kron}(h, a)$, where $\text{kron}(h, a)$ is the matrix in $\text{Mat}_{n \times n}(A)$ with entries $(h_{ij} \cdot a)$.

For any element $a$ of a $C^*$-algebra $A$, the Kronecker product $a \otimes 0$, where $0$ is the $n \times n$ zero matrix over $\mathbb{C}$, is equal to the $n \times n$ zero matrix over $A$.

For any $n \times n$ complex matrix $h \in \text{Mat}_{n \times n}(\mathbb{C})$ and the zero element $0$ of a $C^*$-algebra $A$, the Kronecker product $h \otimes 0$ (the matrix in $\text{Mat}_{n \times n}(A)$ with entries $h_{ij} \cdot 0$) is equal to the zero matrix.

Let $M$ and $N$ be $n \times n$ matrices with entries in $A$. Let $f: A \to B$ be an additive homomorphism from $A$ to an additive commutative monoid $B$. The entrywise application of $f$ to the sum of the matrices $M$ and $N$ is equal to the sum of the matrices obtained by applying $f$ entrywise to $M$ and $N$ individually, expressed as $(M + N).\text{map}(f) = M.\text{map}(f) + N.\text{map}(f)$.

Let $A$ be a $C^*$-algebra. For any $n \times n$ complex matrix $g \in \text{Mat}_{n \times n}(\mathbb{C})$ and any element $b \in A$, let $b \otimes g$ denote the matrix in $\text{Mat}_{n \times n}(A)$ defined by the entrywise product $(b \otimes g)_{ij} = g_{ij} b$. Then the product of the adjoint of the matrix $b \otimes g$ with itself satisfies the identity: $$(b \otimes g)^* (b \otimes g) = (b^* b) \otimes (g^* g)$$ where the multiplication on the left-hand side is matrix multiplication in $\text{Mat}_{n \times n}(A)$, $b^* b$ is the product in $A$, and $g^* g$ is the matrix product of the conjugate transpose of $g$ with $g$ in $\text{Mat}_{n \times n}(\mathbb{C})$.

For a $C^*$-algebra $A$ and a finite set of indices $s$, let $f$ be a collection of $n \times n$ matrices over $A$ indexed by $s$. For any row index $i$ and column index $j$, the $(i, j)$-th entry of the sum of these matrices is equal to the sum of the $(i, j)$-th entries of the individual matrices: $$\left( \sum_{k \in s} f(k) \right)_{i,j} = \sum_{k \in s} f(k)_{i,j}$$

Let $A$ be a unital $C^*$-algebra and $\text{Mat}_{n \times n}(\mathbb{C})$ be the algebra of $n \times n$ complex matrices. Let $\text{selfAdjoint } A$ and $\text{selfAdjoint}(\text{Mat}_{n \times n}(\mathbb{C}))$ denote their respective spaces of self-adjoint (Hermitian) elements. The function $\text{toMatrix}$ is the real-linear map from the tensor product $(\text{selfAdjoint } A) \otimes_{\mathbb{R}} (\text{selfAdjoint}(\text{Mat}_{n \times n}(\mathbb{C})))$ to the space of $n \times n$ matrices over $A$, $\text{Mat}_{n \times n}(A)$, which maps an elementary tensor $a \otimes h$ to the matrix whose $(i, j)$-th entry is $h_{ij} \cdot a$.

Let $A$ be a unital $C^*$-algebra and $\text{selfAdjoint } A$ be its space of self-adjoint elements. Let $\text{selfAdjoint}(\text{Mat}_{n \times n}(\mathbb{C}))$ be the space of $n \times n$ Hermitian matrices. For any $a \in \text{selfAdjoint } A$ and $h \in \text{selfAdjoint}(\text{Mat}_{n \times n}(\mathbb{C}))$, the real-linear map $\text{toMatrix}$ maps the elementary tensor $a \otimes h$ in the composite system $(\text{selfAdjoint } A) \otimes_{\mathbb{R}} (\text{selfAdjoint}(\text{Mat}_{n \times n}(\mathbb{C})))$ to the matrix $\text{kron}(h, a) \in \text{Mat}_{n \times n}(A)$, where the $(i, j)$-th entry of $\text{kron}(h, a)$ is given by $h_{ij} \cdot a$.

Let $A$ be a unital $C^*$-algebra and $\text{Mat}_{n \times n}(\mathbb{C})$ be the algebra of $n \times n$ complex matrices. Let $1_A \in \text{selfAdjoint } A$ be the identity element of $A$ and $I_n \in \text{selfAdjoint}(\text{Mat}_{n \times n}(\mathbb{C}))$ be the $n \times n$ identity matrix. The real-linear map $\text{toMatrix}$ maps the tensor product $1_A \otimes I_n$ to the identity matrix in $\text{Mat}_{n \times n}(A)$, i.e., $\text{toMatrix}(1_A \otimes I_n) = 1_{\text{Mat}_{n \times n}(A)}$.

Let $A$ be a $C^*$-algebra. For any $n \times n$ complex matrix $h \in \text{Mat}_{n \times n}(\mathbb{C})$ and any element $a \in A$, if $h$ is self-adjoint and $a$ is self-adjoint, then the matrix $\text{kron}(h, a) \in \text{Mat}_{n \times n}(A)$ is also self-adjoint, where $\text{kron}(h, a)$ is the matrix with entries $(\text{kron}(h, a))_{ij} = h_{ij} a$.

Let $A$ be a unital $C^*$-algebra and $\text{Mat}_{n \times n}(\mathbb{C})$ be the algebra of $n \times n$ complex matrices. Let $\text{selfAdjoint } A$ and $\text{selfAdjoint}(\text{Mat}_{n \times n}(\mathbb{C}))$ denote their respective spaces of self-adjoint (Hermitian) elements. For any element $z$ in the tensor product $(\text{selfAdjoint } A) \otimes_{\mathbb{R}} (\text{selfAdjoint}(\text{Mat}_{n \times n}(\mathbb{C})))$, the resulting matrix $\text{toMatrix}(z) \in \text{Mat}_{n \times n}(A)$ is self-adjoint.

Let $A$ be a $C^*$-algebra and let $M_n(\mathbb{C})$ denote the set of $n \times n$ complex matrices. For any positive-semidefinite matrix $h \in M_n(\mathbb{C})$ (i.e., $h \ge 0$) and any positive element $a \in A$ (i.e., $a \ge 0$), the matrix in $M_n(A)$ defined by the product $a \otimes h$ (with entries $(h_{ij} a)_{i,j}$) is also positive.

Let $A$ be a unital $C^*$-algebra and $\text{Mat}_n(\mathbb{C})$ be the algebra of $n \times n$ complex matrices. Let $\text{selfAdjoint } A$ and $\text{selfAdjoint}(\text{Mat}_n(\mathbb{C}))$ denote the real vector spaces of self-adjoint elements in their respective algebras. The **matrix cone** is the pointed cone in the composite observable space $(\text{selfAdjoint } A) \otimes_{\mathbb{R}} (\text{selfAdjoint}(\text{Mat}_n(\mathbb{C})))$ defined by the set of elements $z$ whose matrix representation $\text{toMatrix}(z) \in \text{Mat}_n(A)$ is positive (i.e., $\text{toMatrix}(z) \ge 0$).

Let $A$ be a unital $C^*$-algebra and $M_n(\mathbb{C})$ be the algebra of $n \times n$ complex matrices. Let $\text{selfAdjoint } A$ and $\text{selfAdjoint } M_n(\mathbb{C})$ denote the real vector spaces of self-adjoint elements in their respective algebras. For any element $z$ in the composite system $(\text{selfAdjoint } A) \otimes_{\mathbb{R}} (\text{selfAdjoint } M_n(\mathbb{C}))$, $z$ belongs to the matrix cone if and only if its matrix representation $\text{toMatrix}(z) \in M_n(A)$ is positive (i.e., $\text{toMatrix}(z) \ge 0$).

Let $A$ be a unital $C^*$-algebra and let $\text{Mat}_n(\mathbb{C})$ be the algebra of $n \times n$ complex matrices. Consider the composite system formed by the tensor product of their self-adjoint parts, $(\text{selfAdjoint } A) \otimes_{\mathbb{R}} (\text{selfAdjoint}(\text{Mat}_n(\mathbb{C})))$. The minimal tensor cone (the cone generated by elementary tensors $a \otimes h$ where $a \in A$ and $h \in \text{Mat}_n(\mathbb{C})$ are positive) is a subset of the matrix cone (the cone of elements whose matrix representation is positive). That is, $$\text{minTensorCone}(\text{selfAdjoint } A, \text{selfAdjoint}(\text{Mat}_n(\mathbb{C}))) \subseteq \text{matrixCone}(n, A)$$ Consequently, the set of positive matrices in the composite system contains every product of positive observables.

Let $A$ be a unital $C^*$-algebra and $\text{Mat}_n(\mathbb{C})$ be the algebra of $n \times n$ complex matrices. Let $\text{selfAdjoint } A$ and $\text{selfAdjoint}(\text{Mat}_n(\mathbb{C}))$ denote the real vector spaces of self-adjoint elements in their respective algebras. The **matrix cone** $C \subseteq (\text{selfAdjoint } A) \otimes_{\mathbb{R}} (\text{selfAdjoint}(\text{Mat}_n(\mathbb{C})))$—defined as the set of elements $z$ whose matrix representation $\text{toMatrix}(z) \in \text{Mat}_n(A)$ is positive—is **Archimedean**. That is, for any element $z$ in the tensor product space, if $z + \epsilon (1_A \otimes I_n) \in C$ for all $\epsilon > 0$, then $z \in C$, where $1_A$ is the identity of $A$ and $I_n$ is the $n \times n$ identity matrix.

For indices $i, j \in n$, the matrix unit $e_{ij}$ is the $n \times n$ complex matrix whose entry at row $i$ and column $j$ is $1$, and all other entries are $0$.

For any indices $i, j, p, q \in n$, the entry at row $p$ and column $q$ of the matrix unit $e_{ij}$ is equal to $1$ if $i = p$ and $j = q$, and is $0$ otherwise.

For any indices $i$ and $j$, the adjoint of the matrix unit $e_{ij}$ is the matrix unit $e_{ji}$. Using the symbol ${}^*$ to denote the adjoint (conjugate transpose) operation, this is expressed as $(e_{ij})^* = e_{ji}$.

For indices $i, j$, the symmetric unit matrix is the $n \times n$ complex Hermitian matrix defined by $\frac{1}{2}(e_{ij} + e_{ji})$, where $e_{ij}$ represents the matrix unit with $1$ at the $(i, j)$-th entry and $0$ elsewhere.

For indices $i, j \in n$, the asymmetric matrix unit is the $n \times n$ complex self-adjoint (Hermitian) matrix defined by $\frac{i}{2}(e_{ij} - e_{ji})$, where $e_{ij}$ is the standard matrix unit with $1$ at the $(i, j)$-th entry and $0$ elsewhere, and $i$ denotes the imaginary unit $\sqrt{-1}$.

Given an $n \times n$ matrix $M$ with entries in a $C^*$-algebra $A$, this function returns the entry $M_{ij} \in A$ located at row $i$ and column $j$.

Given an $n \times n$ matrix $M$ with entries in a $C^*$-algebra $A$, this function constructs a composite observable in the tensor product space $(\text{selfAdjoint } A) \otimes_{\mathbb{R}} (\text{selfAdjoint } \text{Mat}_{n \times n}(\mathbb{C}))$. The resulting tensor is defined by the sum: $$\sum_{i, j \in n} \left( \text{Re}(M_{ij}) \otimes S_{ij} + \text{Im}(M_{ij}) \otimes A_{ij} \right)$$ where $M_{ij} \in A$ is the $(i, j)$-th entry of the matrix $M$, and $\text{Re}(M_{ij}) = \frac{1}{2}(M_{ij} + M_{ij}^*)$ and $\text{Im}(M_{ij}) = \frac{1}{2i}(M_{ij} - M_{ij}^*)$ are the self-adjoint real and imaginary parts of the entry. Here, $S_{ij} = \frac{1}{2}(e_{ij} + e_{ji})$ denotes the symmetric Hermitian matrix unit and $A_{ij} = \frac{i}{2}(e_{ij} - e_{ji})$ denotes the asymmetric Hermitian matrix unit, where $e_{ij}$ is the standard matrix unit.

Let $A$ be a $C^*$-algebra and $n$ be a finite index set. For any indices $i, j \in n$, let $S_{ij} \in \text{Mat}_{n \times n}(\mathbb{C})$ be the symmetric unit matrix defined by $S_{ij} = \frac{1}{2}(e_{ij} + e_{ji})$, where $e_{ij}$ is the matrix unit with $1$ at the $(i, j)$-th entry and $0$ elsewhere. For any $a \in A$, the $(p, q)$-th entry of the Kronecker product $a \otimes S_{ij} \in \text{Mat}_{n \times n}(A)$ is given by: $$(a \otimes S_{ij})_{pq} = \frac{1}{2} \left( \delta_{ip}\delta_{jq} a + \delta_{jp}\delta_{iq} a \right)$$ where $\delta_{xy}$ is the Kronecker delta, which is $1$ if $x = y$ and $0$ otherwise.

Let $n$ be an index set and $A$ be a $C^*$-algebra. For any indices $i, j, p, q \in n$ and any observable $a \in A$, the $(p, q)$-th entry of the matrix representing the composite observable $a \otimes \text{asymUnit}(i, j)$ is given by $$(a \otimes \text{asymUnit}(i, j))_{pq} = \frac{\mathrm{i}}{2} \cdot \left( \delta_{ip}\delta_{jq} a - \delta_{jp}\delta_{iq} a \right)$$ where $\text{asymUnit}(i, j) = \frac{\mathrm{i}}{2}(e_{ij} - e_{ji})$ is the asymmetric Hermitian matrix unit, $\mathrm{i}$ is the imaginary unit, and $\delta$ denotes the Kronecker delta (which is $1$ if the indices match and $0$ otherwise).

Let $A$ be a unital $C^*$-algebra and $M$ be an $n \times n$ matrix with entries $M_{ij} \in A$. Let $\text{Re}(M_{ij}) = \frac{1}{2}(M_{ij} + M_{ij}^*)$ and $\text{Im}(M_{ij}) = \frac{1}{2i}(M_{ij} - M_{ij}^*)$ be the self-adjoint real and imaginary parts of the entries. Let $S_{ij} = \frac{1}{2}(e_{ij} + e_{ji})$ and $A_{ij} = \frac{i}{2}(e_{ij} - e_{ji})$ be the symmetric and asymmetric Hermitian matrix units in the space of $n \times n$ complex matrices. The matrix representation of the composite observable $\text{ofMatrixSA}(M)$ is given by the sum over all indices $i, j \in n$: $$\text{toMatrix}(\text{ofMatrixSA}(M)) = \sum_{i, j \in n} \left( \text{Re}(M_{ij}) \otimes S_{ij} + \text{Im}(M_{ij}) \otimes A_{ij} \right)$$ where $a \otimes h$ (the operation `kron h a`) denotes the matrix in $\text{Mat}_{n \times n}(A)$ whose entries are the scalar products $h_{pq} \cdot a$.

Let $A$ be a $C^*$-algebra and $M$ be an $n \times n$ matrix with entries in $A$. For any indices $p, q \in n$, the $(p, q)$-th entry of the matrix associated with the self-adjoint composite observable constructed from $M$ is given by $$(\text{toMatrix}(\text{ofMatrixSA}(M)))_{pq} = \frac{1}{2} \left( \text{Re}(M_{pq}) + \text{Re}(M_{qp}) \right) + \frac{i}{2} \left( \text{Im}(M_{pq}) - \text{Im}(M_{qp}) \right)$$ where $\text{Re}(x) = \frac{1}{2}(x + x^*)$ and $\text{Im}(x) = \frac{1}{2i}(x - x^*)$ denote the self-adjoint real and imaginary parts of an element $x \in A$, and $i$ is the imaginary unit.

Let $A$ be a unital $C^*$-algebra. For any $n \times n$ matrix $M$ with entries in $A$ such that $M$ is self-adjoint, there exists a composite observable $z$ in the tensor product space $(\text{selfAdjoint } A) \otimes_{\mathbb{R}} (\text{selfAdjoint } \text{Mat}_{n \times n}(\mathbb{C}))$ such that the matrix representation of $z$ is $M$ (i.e., $\text{toMatrix}(z) = M$).

Given a character $\chi$ in the character space of a $C^*$-algebra $C$, this definition constructs the corresponding positive real-linear functional $\chi|_{sa} : \text{selfAdjoint } C \to \mathbb{R}$. It restricts the complex-linear character $\chi$ to the space of self-adjoint elements (the observables of the system), mapping them to real numbers.

Let $C$ be a $C^*$-algebra and let $\chi: C \to \mathbb{C}$ be a character in the character space of $C$. Let $\chi|_{sa} : \text{selfAdjoint } C \to \mathbb{R}$ be the positive real-linear functional obtained by restricting $\chi$ to the space of self-adjoint elements of $C$. For any $x \in \text{selfAdjoint } C$, the value $\chi|_{sa}(x)$ is equal to $\chi(x)$ when the former is viewed as a complex number.

Let $C$ be a unital $C^*$-algebra and let $\chi$ be a character in the character space of $C$. Let $\chi|_{sa} : \text{selfAdjoint } C \to \mathbb{R}$ be the positive real-linear functional obtained by restricting $\chi$ to the space of self-adjoint elements of $C$. For any composite observable $z$ in the tensor product space $(\text{selfAdjoint } C) \otimes_{\mathbb{R}} (\text{selfAdjoint}(\text{Mat}_{n \times n}(\mathbb{C})))$, it holds that $$(\text{lslice } \chi|_{sa})(z) = \text{toMatrix}(z)_{\chi}$$ where $\text{lslice } \chi|_{sa}$ is the left slice map that applies $\chi|_{sa}$ to the first factor of the tensor product, and $\text{toMatrix}(z)_{\chi}$ is the $n \times n$ complex matrix obtained by applying the character $\chi$ entrywise to the matrix representation $\text{toMatrix}(z) \in \text{Mat}_{n \times n}(C)$.

Let $C$ be a commutative unital $C^*$-algebra. Let $\text{selfAdjoint } C$ and $\text{selfAdjoint } M_n(\mathbb{C})$ denote the spaces of self-adjoint elements of $C$ and the $n \times n$ complex matrix algebra, respectively. For any element $z$ in the composite system $(\text{selfAdjoint } C) \otimes_{\mathbb{R}} (\text{selfAdjoint } M_n(\mathbb{C}))$, if the associated matrix $\text{toMatrix}(z) \in M_{n \times n}(C)$ is positive (i.e., $0 \le \text{toMatrix}(z)$), then $z$ belongs to the maximal tensor cone of the tensor product. Here, the maximal tensor cone consists of all tensors $u$ such that $(\phi_1 \otimes \phi_2)(u) \ge 0$ for every pair of positive linear functionals $\phi_1: \text{selfAdjoint } C \to \mathbb{R}$ and $\phi_2: \text{selfAdjoint } M_n(\mathbb{C}) \to \mathbb{R}$.

Let $A$ and $B$ be unital $C^*$-algebras, and let $f: A \to B$ be a unital positive linear map. Let $z \in (\text{selfAdjoint } A) \otimes_{\mathbb{R}} (\text{selfAdjoint}(\text{Mat}_{n \times n}(\mathbb{C})))$ be a composite observable. Then, applying the map $f$ to the first factor of the composite system and then converting the result into a matrix is equivalent to converting $z$ into its matrix representation in $\text{Mat}_{n \times n}(A)$ and then applying $f$ to each entry of that matrix. That is: $$\text{toMatrix}_B((f \otimes \text{id})(z)) = (\text{toMatrix}_A(z)).\text{map}(f)$$ where $\text{toMatrix}$ is the map sending $a \otimes h$ to the matrix $(h_{ij} a)$.

Let $C$ be a commutative $C^*$-algebra and $B$ be a $C^*$-algebra. Let $f: C \to B$ be a unital positive linear map. For any natural number $k$ and any $k \times k$ matrix $M$ with entries in $C$, if $M$ is positive ($M \ge 0$), then the matrix $M.\text{map } f$ (the matrix obtained by applying $f$ to each entry of $M$) is also positive ($0 \le M.\text{map } f$). This expresses the fact that every unital positive map from a commutative $C^*$-algebra is completely positive.

Let $C$ be a commutative $C^*$-algebra and $B$ be a $C^*$-algebra. Given a unital positive linear map $f : C \to B$, this definition constructs a quantum channel (a completely positive unital map) from $f$. This utilizes the result that for maps originating from a commutative $C^*$-algebra, the property of being unital and positive is sufficient to guarantee complete positivity.

Let $C$ be a commutative $C^*$-algebra and $B$ be a $C^*$-algebra. Let $f : C \to B$ be a unital positive linear map (denoted $f \in C \to_{p1}[\mathbb{C}] B$). For any element $x \in C$, the action of the quantum channel constructed from $f$ (denoted $\text{ofCommutative } f$) on $x$ is equal to $f(x)$. That is, $$(\text{ofCommutative } f)(x) = f(x)$$