PhyslibAlpha.QuantumMechanics.HilbertSpaces.FiniteTarget.Product

Let $\mathcal{H}[\alpha]$ be a finite-dimensional complex Hilbert space indexed by a finite set $\alpha$, equipped with the standard orthonormal basis $\{e_i\}_{i \in \alpha}$. For any linear operator $A: \mathcal{H}[\alpha] \to \mathcal{H}[\alpha]$ and any indices $a, a' \in \alpha$, the $a$-th component of the vector $A e_{a'}$ is equal to the complex inner product $\langle e_a, A e_{a'} \rangle$.

Let $\mathcal{H}_\alpha$ be a finite-dimensional complex Hilbert space indexed by a finite set $\alpha$. For any linear operator $A: \mathcal{H}_\alpha \to \mathcal{H}_\alpha$, any vector $\psi \in \mathcal{H}_\alpha$, and any index $a \in \alpha$, the $a$-th component of the vector $A\psi$ is given by the sum over all $a' \in \alpha$ of the product of the $a$-th component of $A e_{a'}$ and the $a'$-th component of $\psi$: $$(A \psi)_a = \sum_{a' \in \alpha} (A e_{a'})_a \psi_{a'}$$ where $e_{a'}$ is the standard basis vector corresponding to the index $a'$. In operator notation, this corresponds to $(A \psi)_a = \sum_{a'} \langle a|A|a' \rangle \psi_{a'}$.

For a linear operator $A: \mathcal{H}_\alpha \to \mathcal{H}_\alpha$ on a finite-dimensional complex Hilbert space, $\text{onFst}(A)$ is the linear operator on the product Hilbert space $\mathcal{H}_{\alpha \times \beta}$ that acts on the first coordinate. Its action on a state vector $\Psi \in \mathcal{H}_{\alpha \times \beta}$ is defined by $$(\text{onFst}(A) \Psi)(a, b) = \sum_{a' \in \alpha} \langle a | A | a' \rangle \Psi(a', b)$$ where $(a, b) \in \alpha \times \beta$ and $\langle a | A | a' \rangle$ denotes the matrix elements of $A$ with respect to the standard orthonormal basis of $\mathcal{H}_\alpha$.

For a linear operator $B: \mathcal{H}_\beta \to \mathcal{H}_\beta$ acting on a finite-dimensional Hilbert space $\mathcal{H}_\beta$, the definition `onSnd B` is the linear operator on the product Hilbert space $\mathcal{H}_{\alpha \times \beta}$ that acts only on the second coordinate. For any state vector $\Psi \in \mathcal{H}_{\alpha \times \beta}$, the resulting state is defined by the formula: $$((onSnd B) \Psi)(a, b) = \sum_{b' \in \beta} \langle b | B | b' \rangle \Psi(a, b')$$ where $(a, b) \in \alpha \times \beta$, and $\langle b | B | b' \rangle$ denotes the matrix elements of $B$ with respect to the standard orthonormal basis of $\mathcal{H}_\beta$. This corresponds to the lifting of $B$ to the product space as $I_\alpha \otimes B$.

Let $\alpha$ and $\beta$ be finite sets, and let $\mathcal{H}_\alpha$ and $\mathcal{H}_{\alpha \times \beta}$ be the corresponding finite-dimensional complex Hilbert spaces. For any linear operator $A: \mathcal{H}_\alpha \to \mathcal{H}_\alpha$ and any state vector $\Psi \in \mathcal{H}_{\alpha \times \beta}$, the value of the resulting state $(\text{onFst } A) \Psi$ at a configuration $p = (a, b) \in \alpha \times \beta$ is given by: $$((\text{onFst } A) \Psi)(a, b) = \sum_{a' \in \alpha} (A e_{a'})(a) \cdot \Psi(a', b)$$ where $e_{a'}$ denotes the $a'$-th vector of the standard orthonormal basis for $\mathcal{H}_\alpha$, and $(A e_{a'})(a)$ is the $a$-th component of the vector $A e_{a'}$.

Let $\alpha$ and $\beta$ be finite sets, and let $\mathcal{H}_\alpha$ and $\mathcal{H}_\beta$ be the corresponding finite-dimensional complex Hilbert spaces. For any linear operator $B: \mathcal{H}_\beta \to \mathcal{H}_\beta$ and any state vector $\Psi \in \mathcal{H}_{\alpha \times \beta}$, the value of the resulting state $(onSnd B) \Psi$ at a configuration $p = (a, b) \in \alpha \times \beta$ is given by: $$((onSnd B) \Psi)(a, b) = \sum_{b' \in \beta} (B e_{b'})(b) \cdot \Psi(a, b')$$ where $e_{b'}$ denotes the $b'$-th vector of the standard orthonormal basis for $\mathcal{H}_\beta$, and $(B e_{b'})(b)$ is the $b$-th component of the vector $B e_{b'}$.

Let $\alpha$ and $\beta$ be finite sets, and let $\mathcal{H}_\alpha$ and $\mathcal{H}_{\alpha \times \beta}$ be the corresponding finite-dimensional complex Hilbert spaces. For any two linear operators $A, A' : \mathcal{H}_\alpha \to \mathcal{H}_\alpha$, the operator $\text{onFst}$ applied to their composition satisfies: $$\text{onFst}(A \circ A') = \text{onFst}(A) \circ \text{onFst}(A')$$ where $\text{onFst}(A)$ denotes the linear operator on $\mathcal{H}_{\alpha \times \beta}$ that acts as $A$ on the first coordinate and as the identity on the second coordinate.

Let $\alpha$ and $\beta$ be finite sets, and let $\mathcal{H}_\beta$ be the finite-dimensional complex Hilbert space indexed by $\beta$. For any two linear operators $B, B' : \mathcal{H}_\beta \to \mathcal{H}_\beta$, let $\text{onSnd}(B)$ denote the linear operator on the product Hilbert space $\mathcal{H}_{\alpha \times \beta}$ that acts on the second coordinate. Then, acting on the second coordinate with the composition of two operators is equivalent to the composition of the operators acting individually on that coordinate: $$\text{onSnd}(B \circ B') = \text{onSnd}(B) \circ \text{onSnd}(B')$$

Let $\mathcal{H}_\alpha$ and $\mathcal{H}_\beta$ be finite-dimensional complex Hilbert spaces. For any linear operators $A: \mathcal{H}_\alpha \to \mathcal{H}_\alpha$ and $B: \mathcal{H}_\beta \to \mathcal{H}_\beta$, let $\text{onFst}(A)$ and $\text{onSnd}(B)$ be the linear operators on the product Hilbert space $\mathcal{H}_{\alpha \times \beta}$ that act on the first and second coordinates respectively. Then these operators commute, such that: $$\text{onFst}(A) \circ \text{onSnd}(B) = \text{onSnd}(B) \circ \text{onFst}(A)$$

In the finite-dimensional complex Hilbert space $\mathcal{H}[\alpha \times \beta]$ whose coordinates are indexed by the product set $\alpha \times \beta$, the inner product $\langle \Psi, \Phi \rangle$ of two vectors $\Psi$ and $\Phi$ is given by the sum over all indices $p \in \alpha \times \beta$ of the product of the complex conjugate of the component $\Psi(p)$ and the component $\Phi(p)$: $$\langle \Psi, \Phi \rangle = \sum_{p \in \alpha \times \beta} \overline{\Psi(p)} \Phi(p)$$ where $\overline{z}$ denotes the complex conjugate of $z$.

Let $\mathcal{H}[\alpha]$ be a finite-dimensional complex Hilbert space indexed by a finite set $\alpha$, equipped with the standard orthonormal basis $\{|a\rangle\}_{a \in \alpha}$. For any symmetric (Hermitian) linear operator $A$ on $\mathcal{H}[\alpha]$ and any indices $a, a' \in \alpha$, the complex conjugate of the $a'$-th component of the vector $A|a\rangle$ is equal to the $a$-th component of the vector $A|a'\rangle$. In terms of matrix elements, this is expressed as: $$\overline{\langle a' | A | a \rangle} = \langle a | A | a' \rangle$$ where $\overline{z}$ denotes the complex conjugate of $z$, and $\langle a' | A | a \rangle$ denotes the matrix element of $A$ in the standard basis.

Let $\alpha$ and $\beta$ be finite sets, and let $\mathcal{H}[\alpha]$ and $\mathcal{H}[\alpha \times \beta]$ be the corresponding finite-dimensional complex Hilbert spaces. For any linear operator $A: \mathcal{H}[\alpha] \to \mathcal{H}[\alpha]$, if $A$ is symmetric (Hermitian), then the operator $\text{onFst}(A)$ acting on the first coordinate of the product space $\mathcal{H}[\alpha \times \beta]$ is also symmetric.

Let $\alpha$ and $\beta$ be finite sets, and let $\mathcal{H}[\beta]$ and $\mathcal{H}[\alpha \times \beta]$ be the corresponding finite-dimensional complex Hilbert spaces. For any linear operator $B : \mathcal{H}[\beta] \to \mathcal{H}[\beta]$, if $B$ is symmetric (Hermitian), then the operator $\text{onSnd}(B)$ acting on the second coordinate of the product space $\mathcal{H}[\alpha \times \beta]$ is also symmetric.