PhyslibAlpha.QuantumMechanics.HilbertSpaces.FiniteTarget.ProductState

For finite sets $\alpha$ and $\beta$, let $\mathcal{H}_\alpha$ and $\mathcal{H}_ \beta$ be finite-dimensional Hilbert spaces. For vectors $\psi \in \mathcal{H}_\alpha$ and $\phi \in \mathcal{H}_\beta$, the product vector $\psi \otimes \phi$ is a vector in the Hilbert space $\mathcal{H}_{\alpha \times \beta}$ defined such that its components are given by $(\psi \otimes \phi)(a, b) = \psi(a) \phi(b)$ for all $a \in \alpha$ and $b \in \beta$.

For finite sets $\alpha$ and $\beta$, let $\psi \in \mathcal{H}_\alpha$ and $\phi \in \mathcal{H}_\beta$ be vectors in their respective finite-dimensional Hilbert spaces. For any pair $p = (a, b) \in \alpha \times \beta$, the component of the product vector $\psi \otimes \phi$ at index $p$ is given by the product of the individual components: $$(\psi \otimes \phi)(a, b) = \psi(a) \cdot \phi(b)$$

Let $\mathcal{H}_\alpha$ and $\mathcal{H}_\beta$ be finite-dimensional complex Hilbert spaces indexed by finite sets $\alpha$ and $\beta$. For any vectors $\psi, \psi' \in \mathcal{H}_\alpha$ and $\phi, \phi' \in \mathcal{H}_\beta$, the complex inner product of the product vectors $\psi \otimes \phi$ and $\psi' \otimes \phi'$ is the product of their individual inner products: $$\langle \psi \otimes \phi, \psi' \otimes \phi' \rangle = \langle \psi, \psi' \rangle \langle \phi, \phi' \rangle$$

Let $\mathcal{H}_\alpha$ and $\mathcal{H}_\beta$ be finite-dimensional complex Hilbert spaces indexed by finite sets $\alpha$ and $\beta$. For any vectors $\psi \in \mathcal{H}_\alpha$ and $\phi \in \mathcal{H}_\beta$, the norm of their product vector $\psi \otimes \phi$ is equal to the product of their individual norms: $$\|\psi \otimes \phi\| = \|\psi\| \cdot \|\phi\|$$

Let $\mathcal{H}_\alpha$ and $\mathcal{H}_\beta$ be finite-dimensional complex Hilbert spaces indexed by finite sets $\alpha$ and $\beta$. If $\psi \in \mathcal{H}_\alpha$ and $\phi \in \mathcal{H}_\beta$ are unit vectors (that is, $\|\psi\| = 1$ and $\|\phi\| = 1$), then their product vector $\psi \otimes \phi$ is also a unit vector, satisfying $\|\psi \otimes \phi\| = 1$.

Let $\mathcal{H}_\alpha$ and $\mathcal{H}_\beta$ be finite-dimensional complex Hilbert spaces indexed by finite sets $\alpha$ and $\beta$. For any vectors $\psi, \psi' \in \mathcal{H}_\alpha$ and $\phi \in \mathcal{H}_\beta$, the product vector (tensor product) satisfies $(\psi - \psi') \otimes \phi = \psi \otimes \phi - \psi' \otimes \phi$.

Let $\mathcal{H}_\alpha$ and $\mathcal{H}_\beta$ be finite-dimensional complex Hilbert spaces. For any vector $\psi \in \mathcal{H}_\alpha$ and vectors $\phi, \phi' \in \mathcal{H}_\beta$, the product vector $\otimes$ (tensor product) distributes over subtraction in the second argument: $$\psi \otimes (\phi - \phi') = \psi \otimes \phi - \psi \otimes \phi'$$

Let $\mathcal{H}_\alpha$ and $\mathcal{H}_\beta$ be finite-dimensional complex Hilbert spaces indexed by finite sets $\alpha$ and $\beta$. For any real scalar $r \in \mathbb{R}$ and vectors $\psi \in \mathcal{H}_\alpha$ and $\phi \in \mathcal{H}_\beta$, the product vector (tensor product) satisfies $$(r \cdot \psi) \otimes \phi = r \cdot (\psi \otimes \phi)$$ where $\cdot$ denotes scalar multiplication and $\otimes$ denotes the product vector operation.

Let $\mathcal{H}_\alpha$ and $\mathcal{H}_\beta$ be finite-dimensional complex Hilbert spaces. For any real scalar $r \in \mathbb{R}$, and vectors $\psi \in \mathcal{H}_\alpha$ and $\phi \in \mathcal{H}_\beta$, the product vector $\otimes$ (tensor product) satisfies: $$\psi \otimes (r \cdot \phi) = r \cdot (\psi \otimes \phi)$$

Let $\mathcal{H}_\alpha$ and $\mathcal{H}_\beta$ be finite-dimensional complex Hilbert spaces. For any linear operator $A: \mathcal{H}_\alpha \to \mathcal{H}_\alpha$ and vectors $\psi \in \mathcal{H}_\alpha$ and $\phi \in \mathcal{H}_\beta$, the action of the operator $A$ on the first factor of the product vector $\psi \otimes \phi$ is given by $(A\psi) \otimes \phi$. That is, $(A \otimes \mathbb{1})(\psi \otimes \phi) = (A\psi) \otimes \phi$, where $A \otimes \mathbb{1}$ denotes the operator acting on the first coordinate of the product space $\mathcal{H}_{\alpha \times \beta}$.

Let $\mathcal{H}_\alpha$ and $\mathcal{H}_\beta$ be finite-dimensional complex Hilbert spaces. For any linear operator $B: \mathcal{H}_\beta \to \mathcal{H}_\beta$ and any vectors $\psi \in \mathcal{H}_\alpha$ and $\phi \in \mathcal{H}_\beta$, let $I \otimes B$ be the operator on the product space $\mathcal{H}_{\alpha \times \beta}$ that acts only on the second coordinate. Then: $$(I \otimes B)(\psi \otimes \phi) = \psi \otimes (B \phi)$$ where $\psi \otimes \phi$ denotes the product vector in $\mathcal{H}_{\alpha \times \beta}$.

Given an observable $A$ (a self-adjoint linear operator) acting on the finite-dimensional complex Hilbert space $\mathcal{H}_\alpha$, `onFstObservable` defines the corresponding observable on the product Hilbert space $\mathcal{H}_{\alpha \times \beta}$ that acts only on the first subsystem. This represents the operator $A \otimes I$, where $I$ is the identity operator on $\mathcal{H}_\beta$. For a state vector $\Psi \in \mathcal{H}_{\alpha \times \beta}$, the action is given by $$(A \otimes I) \Psi(a, b) = \sum_{a' \in \alpha} \langle a | A | a' \rangle \Psi(a', b)$$ where $(a, b) \in \alpha \times \beta$ and $\langle a | A | a' \rangle$ denotes the matrix elements of $A$ with respect to the standard basis of $\mathcal{H}_\alpha$.

For an observable $B$ acting on the finite-dimensional complex Hilbert space $\mathcal{H}_\beta$, `onSndObservable B` is the observable on the product Hilbert space $\mathcal{H}_{\alpha \times \beta}$ that acts on the second coordinate. Mathematically, this represents the operator $I_\alpha \otimes B$, where $I_\alpha$ is the identity operator on $\mathcal{H}_\alpha$.

Let $\psi \in \mathcal{H}_\alpha$ and $\phi \in \mathcal{H}_\beta$ be unit vectors in finite-dimensional complex Hilbert spaces $\mathcal{H}_\alpha$ and $\mathcal{H}_\beta$. Let $A$ be an observable (a self-adjoint operator) acting on $\mathcal{H}_\alpha$, and let $A \otimes I$ be the corresponding observable acting on the first subsystem of the product space $\mathcal{H}_{\alpha \times \beta}$. The expectation value of $A \otimes I$ in the product state $\psi \otimes \phi$ is equal to the expectation value of $A$ in the state $\psi$: $$\omega_{\psi \otimes \phi}\langle A \otimes I \rangle = \omega_\psi\langle A \rangle$$

Let $\mathcal{H}_\alpha$ and $\mathcal{H}_\beta$ be finite-dimensional complex Hilbert spaces. Let $\psi \in \mathcal{H}_\alpha$ and $\phi \in \mathcal{H}_\beta$ be unit vectors, and let $\psi \otimes \phi \in \mathcal{H}_{\alpha \times \beta}$ be their product vector. For any observable $A$ acting on $\mathcal{H}_\alpha$, let $A \otimes I_\beta$ be the corresponding observable acting on the first coordinate of the product space. The fluctuation vector of $A \otimes I_\beta$ with respect to the state $\psi \otimes \phi$ is given by the tensor product of the fluctuation vector of $A$ with respect to $\psi$ and the vector $\phi$: $$(A \otimes I_\beta)(\psi \otimes \phi) - \langle A \otimes I_\beta \rangle_{\psi \otimes \phi} (\psi \otimes \phi) = (A\psi - \langle A \rangle_\psi \psi) \otimes \phi$$ where $\langle \cdot \rangle_\omega$ denotes the expectation value in the state $\omega$.

Let $\mathcal{H}_\alpha$ and $\mathcal{H}_\beta$ be finite-dimensional complex Hilbert spaces. For any unit vectors $\psi \in \mathcal{H}_\alpha$ and $\phi \in \mathcal{H}_\beta$ (satisfying $\|\psi\| = 1$ and $\|\phi\| = 1$), and any observable $A$ acting on $\mathcal{H}_\alpha$, the variance of the observable $A \otimes I$ (the observable $A$ acting on the first subsystem) in the product state defined by $\psi \otimes \phi$ is equal to the variance of $A$ in the vector state defined by $\psi$: $$\text{Var}_{\psi \otimes \phi}(A \otimes I) = \text{Var}_\psi(A)$$ where $\text{Var}_\omega(X)$ denotes the variance of an observable $X$ with respect to a state $\omega$.

Let $\mathcal{H}_\alpha$ and $\mathcal{H}_\beta$ be finite-dimensional complex Hilbert spaces. Let $\psi \in \mathcal{H}_\alpha$ and $\phi \in \mathcal{H}_\beta$ be unit vectors (i.e., $\|\psi\| = 1$ and $\|\phi\| = 1$), and let $\psi \otimes \phi$ be their product vector in the tensor product space $\mathcal{H}_{\alpha \times \beta}$. For any observables $a, b$ acting on $\mathcal{H}_\alpha$, let $a \otimes I$ and $b \otimes I$ be the corresponding observables acting on the first coordinate of the product space. The covariance of $a \otimes I$ and $b \otimes I$ with respect to the product state $\psi \otimes \phi$ is equal to the covariance of $a$ and $b$ with respect to the state $\psi$: $$\text{Cov}_{\psi \otimes \phi}(a \otimes I, b \otimes I) = \text{Cov}_\psi(a, b)$$

Let $\mathcal{H}_\alpha$ and $\mathcal{H}_\beta$ be finite-dimensional complex Hilbert spaces. Let $\psi \in \mathcal{H}_\alpha$ and $\phi \in \mathcal{H}_\beta$ be unit vectors, and let $\omega_\psi$ and $\omega_{\psi \otimes \phi}$ be the vector states associated with $\psi$ and the product vector $\psi \otimes \phi$ respectively (where $\omega_\psi(x) = \langle \psi, x \psi \rangle$). For any two observables $a, b$ acting on $\mathcal{H}_\alpha$, let $a \otimes I$ and $b \otimes I$ be the corresponding observables acting on the first coordinate of the product Hilbert space $\mathcal{H}_{\alpha \times \beta}$. Then the centered Gram defect of $a \otimes I$ and $b \otimes I$ in the product state $\omega_{\psi \otimes \phi}$ is equal to the centered Gram defect of $a$ and $b$ in the state $\omega_\psi$: $$\text{centeredGramDefect}_{\omega_{\psi \otimes \phi}}(a \otimes I, b \otimes I) = \text{centeredGramDefect}_{\omega_\psi}(a, b)$$

Let $\mathcal{H}_\alpha$ and $\mathcal{H}_\beta$ be finite-dimensional complex Hilbert spaces. Given unit vectors $\psi \in \mathcal{H}_\alpha$ and $\phi \in \mathcal{H}_\beta$, the expectation value of an observable $B$ acting on the second coordinate of the product space $\mathcal{H}_{\alpha \times \beta}$ with respect to the product state $\psi \otimes \phi$ is equal to the expectation value of $B$ with respect to the factor state $\phi$. Mathematically, $$\omega_{\psi \otimes \phi}\langle I \otimes B \rangle = \omega_\phi\langle B \rangle$$ where $\omega_v$ denotes the vector state associated with a unit vector $v$ (defined by $\omega_v\langle X \rangle = \langle v, Xv \rangle$), and $I \otimes B$ represents the observable acting as the identity on $\mathcal{H}_\alpha$ and as $B$ on $\mathcal{H}_\beta$.

Let $\mathcal{H}_\alpha$ and $\mathcal{H}_\beta$ be finite-dimensional complex Hilbert spaces. Let $\psi \in \mathcal{H}_\alpha$ and $\phi \in \mathcal{H}_\beta$ be unit vectors, and let $\psi \otimes \phi$ be their product vector in $\mathcal{H}_{\alpha \times \beta}$. For any observable $B$ acting on $\mathcal{H}_\beta$, let $I_\alpha \otimes B$ be the corresponding observable acting on the second coordinate of the product space. The fluctuation vector of $\psi \otimes \phi$ under $I_\alpha \otimes B$ is given by the product of $\psi$ and the fluctuation vector of $\phi$ under $B$: $$(I_\alpha \otimes B)(\psi \otimes \phi) - \langle I_\alpha \otimes B \rangle_{\psi \otimes \phi} (\psi \otimes \phi) = \psi \otimes (B\phi - \langle B \rangle_\phi \phi)$$ where $\langle A \rangle_v$ denotes the expectation value of the operator $A$ in the vector state defined by $v$.

Let $\mathcal{H}_\alpha$ and $\mathcal{H}_\beta$ be finite-dimensional complex Hilbert spaces, and let $\psi \in \mathcal{H}_\alpha$ and $\phi \in \mathcal{H}_\beta$ be unit vectors (i.e., $\|\psi\| = 1$ and $\|\phi\| = 1$). For any observable $B$ acting on $\mathcal{H}_\beta$, let $I_\alpha \otimes B$ denote the observable acting on the second coordinate of the product Hilbert space $\mathcal{H}_{\alpha \times \beta}$. Then the variance of $I_\alpha \otimes B$ with respect to the product vector state $\psi \otimes \phi$ is equal to the variance of $B$ with respect to the state $\phi$: $$\text{Var}_{\psi \otimes \phi}(I_\alpha \otimes B) = \text{Var}_\phi(B)$$

Let $\mathcal{H}_\alpha$ and $\mathcal{H}_\beta$ be finite-dimensional complex Hilbert spaces, and let $\psi \in \mathcal{H}_\alpha$ and $\phi \in \mathcal{H}_\beta$ be unit vectors. Let $\omega_{\psi \otimes \phi}$ denote the vector state associated with the product vector $\psi \otimes \phi$ on the product space, and let $\omega_\phi$ denote the vector state associated with $\phi$ on $\mathcal{H}_\beta$. For any two observables $a$ and $b$ acting on $\mathcal{H}_\beta$, the covariance of their extensions to the second coordinate of the product space, $I_\alpha \otimes a$ and $I_\alpha \otimes b$, is equal to the covariance of $a$ and $b$ in the state $\omega_\phi$: $$\text{Cov}_{\omega_{\psi \otimes \phi}}(I_\alpha \otimes a, I_\alpha \otimes b) = \text{Cov}_{\omega_\phi}(a, b)$$

Let $\mathcal{H}_\alpha$ and $\mathcal{H}_\beta$ be finite-dimensional complex Hilbert spaces. Let $\psi \in \mathcal{H}_\alpha$ and $\phi \in \mathcal{H}_\beta$ be unit vectors (i.e., $\|\psi\| = 1$ and $\|\phi\| = 1$), and let $\omega_{\psi \otimes \phi}$ denote the vector state associated with the product vector $\psi \otimes \phi$ on the product Hilbert space $\mathcal{H}_{\alpha \times \beta}$. For any two observables $a, b$ acting on $\mathcal{H}_\beta$, let $I \otimes a$ and $I \otimes b$ be the corresponding observables acting only on the second coordinate of the product space. Then the centered Gram defect of $I \otimes a$ and $I \otimes b$ with respect to the state $\omega_{\psi \otimes \phi}$ is equal to the centered Gram defect of $a$ and $b$ with respect to the state $\omega_\phi$ (the vector state associated with $\phi$ on $\mathcal{H}_\beta$): $$\text{centeredGramDefect}_{\omega_{\psi \otimes \phi}}(I \otimes a, I \otimes b) = \text{centeredGramDefect}_{\omega_\phi}(a, b)$$

Let $\mathcal{H}$ be a finite-dimensional complex Hilbert space. For any unit vector $\chi \in \mathcal{H}$ and any observable $c$ (a self-adjoint operator on $\mathcal{H}$), let $\langle c \rangle_\chi = \langle \chi, c \chi \rangle$ denote the expectation value of $c$ in the state $\chi$. The fluctuation vector is defined as $c \chi - \langle c \rangle_\chi \chi$. Then the inner product of $\chi$ with the fluctuation vector is zero, meaning they are orthogonal: $$\langle \chi, c \chi - \langle c \rangle_\chi \chi \rangle = 0$$

Let $\mathcal{H}_\alpha$ and $\mathcal{H}_\beta$ be finite-dimensional complex Hilbert spaces. Let $\psi \in \mathcal{H}_\alpha$ and $\phi \in \mathcal{H}_\beta$ be unit vectors (i.e., $\|\psi\| = 1$ and $\|\phi\| = 1$). Let $\omega_{\psi \otimes \phi}$ denote the vector state associated with the product vector $\psi \otimes \phi$ on the product Hilbert space $\mathcal{H}_{\alpha \times \beta}$. For any observable $a$ acting on $\mathcal{H}_\alpha$ (with $a \otimes I$ being its extension to the product space) and any observable $b$ acting on $\mathcal{H}_\beta$ (with $I \otimes b$ being its extension), the covariance between these two observables in the state $\omega_{\psi \otimes \phi}$ is zero: $$\text{Cov}_{\omega_{\psi \otimes \phi}}(a \otimes I, I \otimes b) = 0$$