PhyslibAlpha.ProbabilisticTheory.CStarAlgebra.Stinespring.Kernel

Given a Hilbert space $H$ and a natural number $n$, let $M$ be an $n \times n$ matrix whose entries $M_{ij}$ are bounded linear operators on $H$ (i.e., $M_{ij} \in \mathcal{B}(H)$). The function `blockMatrixMap` defines a bounded linear operator on the direct sum space $H^n \cong \bigoplus_{k=1}^n H$ such that for any vector $\mathbf{x} = (x_1, \dots, x_n) \in H^n$, the $i$-th component of the output is given by $\sum_{j=1}^n M_{ij} x_j$.

Let $H$ be a Hilbert space and $n$ be a natural number. Let $M$ be an $n \times n$ matrix of bounded linear operators on $H$, where $M_{ij} \in \mathcal{B}(H)$ denotes the entry in the $i$-th row and $j$-th column. For any vector $\mathbf{x} = (x_1, \dots, x_n)$ in the direct sum space $H^n \cong \bigoplus_{k=1}^n H$, the $i$-th component of the vector produced by the operator $\text{blockMatrixMap}(M)$ acting on $\mathbf{x}$ is given by the sum: $$((\text{blockMatrixMap} \, M) \mathbf{x})_i = \sum_{j=1}^n M_{ij} x_j$$ where $(\cdot)_i$ denotes the projection onto the $i$-th coordinate.

Let $H$ be a Hilbert space and $n$ be a natural number. Let $M$ be an $n \times n$ matrix whose entries $M_{ij}$ are bounded linear operators on $H$ (i.e., $M_{ij} \in \mathcal{B}(H)$). Let $M^*$ denote the adjoint matrix, whose entries are given by $(M^*)_{ij} = (M_{ji})^*$, where $(M_{ji})^*$ is the adjoint operator of $M_{ji}$. Then the bounded linear operator on the direct sum space $H^n \cong \bigoplus_{k=1}^n H$ induced by the matrix $M^*$ is equal to the adjoint of the bounded linear operator induced by $M$. That is, $$\text{blockMatrixMap}(M^*) = (\text{blockMatrixMap}(M))^*$$

Let $H$ be a Hilbert space and $n$ be a natural number. For any two $n \times n$ matrices $M$ and $N$ whose entries are bounded linear operators on $H$ (i.e., $M_{ij}, N_{ij} \in \mathcal{B}(H)$), the bounded linear operator on the direct sum space $H^n$ induced by the matrix product $M \cdot N$ is equal to the composition of the bounded linear operators induced by $M$ and $N$ individually. That is, $$\text{blockMatrixMap}(M \cdot N) = \text{blockMatrixMap}(M) \circ \text{blockMatrixMap}(N)$$ where $\text{blockMatrixMap}$ is the map from operator-valued matrices to operators on $H^n$.

Let $H$ be a Hilbert space and $n$ be a natural number. Let $1$ denote the $n \times n$ identity matrix whose entries are bounded linear operators on $H$ (where the diagonal entries are the identity operator $\text{id}_H$ and the off-diagonal entries are the zero operator). The bounded linear operator on the direct sum space $H^n \cong \bigoplus_{k=1}^n H$ induced by this matrix is the identity operator on $H^n$. That is, $$\text{blockMatrixMap}(1) = \text{id}_{H^n}$$ where $\text{blockMatrixMap}$ is the map from operator-valued matrices to operators on $H^n$.

Let $H$ be a Hilbert space and $n$ be a natural number. For any two $n \times n$ matrices $M$ and $N$ whose entries are bounded linear operators on $H$ (i.e., $M_{ij}, N_{ij} \in \mathcal{B}(H)$), the bounded linear operator on the direct sum space $H^n \cong \bigoplus_{k=1}^n H$ induced by the matrix sum $M + N$ is equal to the sum of the bounded linear operators induced by $M$ and $N$ individually. That is, $$\text{blockMatrixMap}(M + N) = \text{blockMatrixMap}(M) + \text{blockMatrixMap}(N)$$ where $\text{blockMatrixMap}$ is the map from operator-valued matrices to operators on $H^n$.

For any natural number $n$ and Hilbert space $H$, the bounded linear operator on $H^n \cong \bigoplus_{k=1}^n H$ induced by the $n \times n$ zero matrix of operators (where each entry is the zero operator on $H$) is the zero operator on $H^n$.

For a natural number $n$ and a complex Hilbert space $H$, `blockMatrixRepresentation` is the unital $\mathbb{C}$-star-algebra homomorphism from the algebra of $n \times n$ matrices with entries in $\mathcal{B}(H)$ (the bounded linear operators on $H$) to the algebra of bounded linear operators on the direct sum Hilbert space $H^n \cong \bigoplus_{k=1}^n H$. This homomorphism maps an operator-valued matrix $M$ to the operator that acts on a vector $\mathbf{x} = (x_1, \dots, x_n) \in H^n$ such that its $i$-th component is $\sum_{j=1}^n M_{ij} x_j$. It preserves the additive, multiplicative, scalar multiplication, identity, and adjoint (star) structures of the matrix algebra.

Let $H$ be a complex Hilbert space and $n$ be a natural number. Let $M$ be an $n \times n$ matrix where each entry $M_{ij}$ is a bounded linear operator on $H$. If $M$ is a positive operator-valued matrix ($M \ge 0$), then the bounded linear operator on the direct sum Hilbert space $H^n \cong \bigoplus_{k=1}^n H$ induced by $M$ (via `blockMatrixMap`) is a positive operator.

Let $H$ be a complex Hilbert space and let $M$ be an $n \times n$ matrix where each entry $M_{ij}$ is a bounded linear operator on $H$. If $M$ is a positive operator-valued matrix ($M \ge 0$), then for any vector $x = (x_1, \dots, x_n)$ in the direct sum Hilbert space $H^n = \bigoplus_{i=1}^n H$, the following inequality holds: $$\sum_{i=1}^n \sum_{j=1}^n \langle x_i, M_{ij} x_j \rangle \ge 0$$ where $\langle \cdot, \cdot \rangle$ denotes the inner product on $H$.

Let $A$ and $B$ be $C^*$-algebras. For any completely positive map $J : A \to B$ and any element $a \in A$, the map $J$ preserves the star operation, satisfying $J(a^*) = J(a)^*$.

Given a sequence of $n$ elements $a = (a_0, a_1, \dots, a_{n-1})$ in an algebra $A$ (typically a $C^*$-algebra), where $n$ is a positive integer, the function defines an $n \times n$ matrix $M$ whose entries $M_{ij}$ are given by $a_j$ if $i = 0$ and $0$ otherwise. In other words, $M$ is a square matrix where the zeroth row is the vector $a$ and all other rows consist of zero elements.

Given a positive integer $n$ and a sequence of $n$ elements $a = (a_0, a_1, \dots, a_{n-1})$ in a $C^*$-algebra $A$, the Gram matrix of $a$ is the $n \times n$ matrix over $A$ whose entry at row $i$ and column $j$ is defined as $a_i^* a_j$, where $a_i^*$ denotes the adjoint of $a_i$.

For a positive integer $n$ and a sequence of elements $a = (a_0, a_1, \dots, a_{n-1})$ in a $C^*$-algebra $A$, the entry of the Gram matrix of $a$ at row $i$ and column $j$ is given by $a_i^* a_j$, where $a_i^*$ denotes the adjoint of $a_i$.

Let $A$ be a unital $C^*$-algebra. For any positive integer $n$ and any sequence of elements $a_0, a_1, \dots, a_{n-1} \in A$, the Gram matrix $G \in M_n(A)$ with entries $G_{ij} = a_i^* a_j$ (where $a_i^*$ denotes the adjoint of $a_i$) is positive, i.e., $G \ge 0$.

Let $A$ be a unital $C^*$-algebra and $H$ be a complex Hilbert space. Let $J: A \to \mathcal{B}(H)$ be a completely positive map, where $\mathcal{B}(H)$ denotes the space of bounded linear operators on $H$. For any positive integer $n$ and any sequence of elements $a_0, a_1, \dots, a_{n-1} \in A$, let $G$ be the $n \times n$ Gram matrix with entries $G_{ij} = a_i^* a_j$. Then the matrix $J(G)$, defined by applying $J$ to each entry of $G$ such that $(J(G))_{ij} = J(a_i^* a_j)$, is a positive operator on the direct sum Hilbert space $H^n$.

Let $A$ be a unital $C^*$-algebra and $H$ be a complex Hilbert space. For any completely positive map $J: A \to \mathcal{B}(H)$, any finite sequence of elements $a_1, \dots, a_n \in A$ (where $n > 0$), and any sequence of vectors $x_1, \dots, x_n \in H$, the following inequality holds: $$\sum_{i=1}^n \sum_{j=1}^n \langle J(a_i^* a_j) x_j, x_i \rangle \ge 0$$ where $\mathcal{B}(H)$ denotes the space of bounded linear operators on $H$, $a_i^*$ is the adjoint of $a_i$, and $\langle \cdot, \cdot \rangle$ is the inner product on $H$.

Let $A$ be a unital $C^*$-algebra and $H$ be a complex Hilbert space. Let $J: A \to \mathcal{B}(H)$ be a completely positive map, where $\mathcal{B}(H)$ denotes the space of bounded linear operators on $H$. For any $n \in \mathbb{N}$, any finite collection of elements $a_1, \dots, a_n \in A$, and any vectors $x_1, \dots, x_n \in H$, the following inequality holds: $$\sum_{i, j=1}^n \langle J(a_i^* a_j) x_j, x_i \rangle \ge 0$$ where $\langle \cdot, \cdot \rangle$ denotes the inner product on $H$ and $a_i^*$ is the adjoint of $a_i$ in $A$.

Let $A$ be a unital $C^*$-algebra and $H$ be a complex Hilbert space. Let $J : A \to \mathcal{B}(H)$ be a completely positive map, where $\mathcal{B}(H)$ denotes the algebra of bounded linear operators on $H$. For any positive integer $n$, any sequence of elements $a_0, \dots, a_{n-1} \in A$, and any sequence of vectors $x_0, \dots, x_{n-1} \in H$, the following inequality holds: $$0 \leq \sum_{i,j=0}^{n-1} \langle x_i, J(a_i^* a_j) x_j \rangle$$ where $\langle \cdot, \cdot \rangle$ denotes the inner product on $H$, which is linear in the second argument.

Let $A$ be a unital $C^*$-algebra and $H$ be a complex Hilbert space. Let $\mathcal{B}(H)$ denote the space of bounded linear operators on $H$. If $J: A \to \mathcal{B}(H)$ is a completely positive map, then for any natural number $n$, any collection of elements $a_0, \dots, a_{n-1} \in A$, and any collection of vectors $x_0, \dots, x_{n-1} \in H$, the following inequality holds: $$\sum_{i,j=0}^{n-1} \langle x_i, J(a_i^* a_j) x_j \rangle \ge 0$$ where $\langle \cdot, \cdot \rangle$ denotes the inner product on $H$.

Let $A$ be a unital $C^*$-algebra and $H$ be a complex Hilbert space. Let $J : A \to \mathcal{B}(H)$ be a completely positive map, where $\mathcal{B}(H)$ denotes the algebra of bounded linear operators on $H$. Given a Stinespring witness $W$ for $J$—consisting of a representation $\pi : A \to \mathcal{B}(K)$ on a Hilbert space $K$ and a bounded linear operator $V : H \to K$—then for any $a \in A$, the following equality holds: $$J(a) = V^* \pi(a) V$$ where $V^*$ is the adjoint operator of $V$.