QuantumInfo.ForMathlib.HayataGroup.TraceInequality.BlockDiagonal

Given a complex Hilbert space $\mathcal{H}$, the two-fold Hilbert sum $HSum(\mathcal{H})$ is the $\ell^2$ direct sum of two copies of $\mathcal{H}$, denoted as $\mathcal{H} \oplus \mathcal{H}$. It is defined as the space of pairs $(x_0, x_1)$ with $x_i \in \mathcal{H}$ equipped with the $\ell^2$ norm $\|(x_0, x_1)\| = \sqrt{\|x_0\|^2 + \|x_1\|^2}$.

For a complex Hilbert space $\mathcal{H}$, let $HSum(\mathcal{H})$ be the two-fold Hilbert sum (the $\ell^2$ direct sum $\mathcal{H} \oplus \mathcal{H}$ with norm $\|(x_0, x_1)\| = \sqrt{\|x_0\|^2 + \|x_1\|^2}$). This definition is the continuous linear equivalence between $HSum(\mathcal{H})$ and the function space $\text{Fin } 2 \to \mathcal{H}$, which identifies the $\ell^2$ direct sum with the plain product space of two copies of $\mathcal{H}$.

Given a complex Hilbert space $\mathcal{H}$, let $HSum(\mathcal{H}) = \mathcal{H} \oplus \mathcal{H}$ be the two-fold Hilbert sum. For each index $i \in \{0, 1\}$, this function is the continuous linear map $HSum(\mathcal{H}) \to \mathcal{H}$ that projects an element of the sum onto its $i$-th coordinate.

For a complex Hilbert space $\mathcal{H}$ and an index $i \in \{0, 1\}$, this is the continuous linear map $\iota_i : \mathcal{H} \to HSum(\mathcal{H})$ that embeds an element $x \in \mathcal{H}$ into the $i$-th summand of the two-fold Hilbert sum $\mathcal{H} \oplus \mathcal{H}$. Specifically, $\iota_0(x) = (x, 0)$ and $\iota_1(x) = (0, x)$.

Let $\mathcal{H}$ be a complex Hilbert space and let $\mathcal{H} \oplus \mathcal{H}$ be the two-fold Hilbert sum. For any indices $i, j \in \{0, 1\}$ and any vector $x \in \mathcal{H}$, let $\iota_j : \mathcal{H} \to \mathcal{H} \oplus \mathcal{H}$ be the $j$-th inclusion map and $\pi_i : \mathcal{H} \oplus \mathcal{H} \to \mathcal{H}$ be the $i$-th coordinate projection. The composition of these maps satisfies: $$\pi_i(\iota_j(x)) = \begin{cases} x & \text{if } i = j \\ 0 & \text{if } i \neq j \end{cases}$$

Let $\mathcal{H}$ be a complex Hilbert space. For any indices $i, j \in \{0, 1\}$ and any vectors $x, y \in \mathcal{H}$, the inner product between the inclusions $\iota_i(x)$ and $\iota_j(y)$ in the two-fold Hilbert sum $\mathcal{H} \oplus \mathcal{H}$ is given by $\langle \iota_i(x), \iota_j(y) \rangle = \langle x, y \rangle$ if $i = j$, and $\langle \iota_i(x), \iota_j(y) \rangle = 0$ if $i \neq j$.

For a complex Hilbert space $\mathcal{H}$ and an index $i \in \{0, 1\}$, the adjoint of the $i$-th inclusion map $\iota_i : \mathcal{H} \to \mathcal{H} \oplus \mathcal{H}$ into the two-fold Hilbert sum is the $i$-th coordinate projection $\pi_i : \mathcal{H} \oplus \mathcal{H} \to \mathcal{H}$. That is, $\iota_i^* = \pi_i$.

Let $\mathcal{H}$ be a complex Hilbert space and let $\mathcal{H} \oplus \mathcal{H}$ be the two-fold Hilbert sum. For each index $i \in \{0, 1\}$, let $\pi_i : \mathcal{H} \oplus \mathcal{H} \to \mathcal{H}$ be the $i$-th coordinate projection and $\iota_i : \mathcal{H} \to \mathcal{H} \oplus \mathcal{H}$ be the $i$-th inclusion map. Then the adjoint of the projection map $\pi_i$ is the inclusion map $\iota_i$, i.e., $\pi_i^* = \iota_i$.

Let $\mathcal{H}$ be a complex Hilbert space and let $HSum(\mathcal{H}) = \mathcal{H} \oplus \mathcal{H}$ be the $\ell^2$ direct sum of two copies of $\mathcal{H}$. For any element $z \in \mathcal{H} \oplus \mathcal{H}$, the sum of its coordinate projections mapped back into the Hilbert sum via the inclusion maps equals the original vector: $$\iota_0(p_0(z)) + \iota_1(p_1(z)) = z,$$ where $p_i : \mathcal{H} \oplus \mathcal{H} \to \mathcal{H}$ is the $i$-th coordinate projection and $\iota_i : \mathcal{H} \to \mathcal{H} \oplus \mathcal{H}$ is the $i$-th inclusion map for $i \in \{0, 1\}$.

Given two bounded linear operators $A, B \in \mathcal{B}(\mathcal{H})$ on a complex Hilbert space $\mathcal{H}$, the block diagonal operator $\text{diag}(A, B)$ is the bounded linear operator on the two-fold Hilbert sum $\mathcal{H} \oplus \mathcal{H}$ defined by the expression $\iota_0 \circ A \circ \pi_0 + \iota_1 \circ B \circ \pi_1$. Here, $\pi_i : \mathcal{H} \oplus \mathcal{H} \to \mathcal{H}$ are the coordinate projections and $\iota_i : \mathcal{H} \to \mathcal{H} \oplus \mathcal{H}$ are the coordinate inclusions for $i \in \{0, 1\}$. Effectively, this operator maps a vector $(x_0, x_1) \in \mathcal{H} \oplus \mathcal{H}$ to $(Ax_0, Bx_1)$.

Given four bounded linear operators $A_{00}, A_{01}, A_{10}, A_{11} \in \mathcal{B}(\mathcal{H})$ on a complex Hilbert space $\mathcal{H}$, this definition constructs a $2 \times 2$ block operator on the two-fold Hilbert sum $\mathcal{H} \oplus \mathcal{H}$. The resulting operator corresponds to the matrix representation $$\begin{pmatrix} A_{00} & A_{01} \\ A_{10} & A_{11} \end{pmatrix}$$ and is formally defined as the sum of compositions $\sum_{i,j \in \{0, 1\}} \iota_i \circ A_{ij} \circ \pi_j$, where $\iota_i: \mathcal{H} \to \mathcal{H} \oplus \mathcal{H}$ and $\pi_j: \mathcal{H} \oplus \mathcal{H} \to \mathcal{H}$ are the canonical inclusion and projection maps for the $i$-th and $j$-th coordinates, respectively.

Let $\mathcal{H}$ be a complex Hilbert space and let $HSum(\mathcal{H}) = \mathcal{H} \oplus \mathcal{H}$ be the two-fold Hilbert sum. Let $\pi_i : HSum(\mathcal{H}) \to \mathcal{H}$ for $i \in \{0, 1\}$ denote the coordinate projections. For any two bounded linear operators $T, S \in \mathcal{B}(HSum(\mathcal{H}))$, if for every $z \in HSum(\mathcal{H})$ the projections of the results are equal, i.e., $\pi_0(Tz) = \pi_0(Sz)$ and $\pi_1(Tz) = \pi_1(Sz)$, then $T = S$.

Let $\mathcal{H}$ be a complex Hilbert space and let $A, B \in \mathcal{B}(\mathcal{H})$ be bounded linear operators. Then the adjoint of the block diagonal operator $\text{diag}(A, B)$ on the two-fold Hilbert sum $\mathcal{H} \oplus \mathcal{H}$ is given by the block diagonal operator of the adjoints: $$\text{diag}(A, B)^* = \text{diag}(A^*, B^*)$$

Let $\mathcal{H}$ be a complex Hilbert space and $\mathcal{B}(\mathcal{H})$ be the algebra of bounded linear operators on $\mathcal{H}$. Let $\mathcal{H} \oplus \mathcal{H}$ denote the two-fold Hilbert sum. The mapping `blockDiagonalHom` is the $\mathbb{R}$-$\star$-algebra homomorphism from the product algebra $\mathcal{B}(\mathcal{H}) \times \mathcal{B}(\mathcal{H})$ to $\mathcal{B}(\mathcal{H} \oplus \mathcal{H})$ that sends a pair of operators $(A, B)$ to the block-diagonal operator $\text{diag}(A, B)$.

Let $\mathcal{H}$ be a complex Hilbert space and let $\mathcal{B}(\mathcal{H})$ be the algebra of bounded linear operators on $\mathcal{H}$. For any pair of operators $p = (A, B) \in \mathcal{B}(\mathcal{H}) \times \mathcal{B}(\mathcal{H})$, the value of the $\mathbb{R}$-$\star$-algebra homomorphism `blockDiagonalHom` applied to $p$ is the block diagonal operator $\text{diag}(A, B)$ on the two-fold Hilbert sum $\mathcal{H} \oplus \mathcal{H}$. That is, $$\text{blockDiagonalHom}(A, B) = \text{diag}(A, B).$$

Let $\mathcal{H}$ be a complex Hilbert space and $A, B \in \mathcal{B}(\mathcal{H})$ be bounded linear operators. For any vector $z \in \mathcal{H} \oplus \mathcal{H}$, let $\pi_0 : \mathcal{H} \oplus \mathcal{H} \to \mathcal{H}$ be the projection onto the first coordinate. Then the application of the block diagonal operator $\text{diag}(A, B)$ followed by the projection $\pi_0$ satisfies $$\pi_0(\text{diag}(A, B) z) = A(\pi_0 z).$$

Let $\mathcal{H}$ be a complex Hilbert space and $\mathcal{H} \oplus \mathcal{H}$ be the two-fold Hilbert sum. For any bounded linear operators $A, B \in \mathcal{B}(\mathcal{H})$ and any vector $z \in \mathcal{H} \oplus \mathcal{H}$, the projection of the action of the block diagonal operator $\text{diag}(A, B)$ onto the second coordinate is equal to $B$ applied to the second coordinate of $z$. That is, $$\pi_1(\text{diag}(A, B)z) = B(\pi_1(z))$$ where $\pi_1 : \mathcal{H} \oplus \mathcal{H} \to \mathcal{H}$ is the projection onto the second coordinate.

For a complex Hilbert space $\mathcal{H}$, the block diagonal operator $\text{diag}(I, I)$ consisting of identity operators $I \in \mathcal{B}(\mathcal{H})$ is equal to the identity operator on the two-fold Hilbert sum $\mathcal{H} \oplus \mathcal{H}$.

Let $\mathcal{H}$ be a complex Hilbert space and let $A, B \in \mathcal{B}(\mathcal{H})$ be bounded linear operators. If $A$ and $B$ are non-negative (i.e., $0 \leq A$ and $0 \leq B$), then the block diagonal operator $\text{diag}(A, B)$ on the two-fold Hilbert sum $\mathcal{H} \oplus \mathcal{H}$ is also non-negative, $0 \leq \text{diag}(A, B)$.

Let $\mathcal{H}$ be a complex Hilbert space and $\mathcal{B}(\mathcal{H})$ be the space of bounded linear operators on $\mathcal{H}$. For any four operators $A_{00}, A_{01}, A_{10}, A_{11} \in \mathcal{B}(\mathcal{H})$ and any vector $z = (z_0, z_1) \in \mathcal{H} \oplus \mathcal{H}$, the first coordinate (index 0) of the vector resulting from the application of the $2 \times 2$ block operator $$\begin{pmatrix} A_{00} & A_{01} \\ A_{10} & A_{11} \end{pmatrix}$$ to $z$ is given by $A_{00} z_0 + A_{01} z_1$.

Let $\mathcal{H}$ be a complex Hilbert space and let $\mathcal{B}(\mathcal{H})$ be the space of bounded linear operators on $\mathcal{H}$. For any four operators $A_{00}, A_{01}, A_{10}, A_{11} \in \mathcal{B}(\mathcal{H})$, let $$T = \begin{pmatrix} A_{00} & A_{01} \\ A_{10} & A_{11} \end{pmatrix}$$ be the $2 \times 2$ block operator acting on the Hilbert sum $\mathcal{H} \oplus \mathcal{H}$. For any vector $z \in \mathcal{H} \oplus \mathcal{H}$, let $\pi_0(z)$ and $\pi_1(z)$ denote the projections onto the first and second coordinates, respectively. Then the second coordinate of the vector $Tz$ is given by $$\pi_1(Tz) = A_{10}(\pi_0(z)) + A_{11}(\pi_1(z)).$$

Let $\mathcal{H}$ be a complex Hilbert space and let $A_{00}, A_{01}, A_{10}, A_{11} \in \mathcal{B}(\mathcal{H})$ be bounded linear operators. The adjoint of the $2 \times 2$ block operator $\begin{pmatrix} A_{00} & A_{01} \\ A_{10} & A_{11} \end{pmatrix}$ acting on the Hilbert sum $\mathcal{H} \oplus \mathcal{H}$ is given by the block operator $$\begin{pmatrix} A_{00} & A_{01} \\ A_{10} & A_{11} \end{pmatrix}^* = \begin{pmatrix} A_{00}^* & A_{10}^* \\ A_{01}^* & A_{11}^* \end{pmatrix},$$ where $A^*$ denotes the adjoint of the operator $A$.