PhyslibAlpha.ProbabilisticTheory.CStarAlgebra.Uncertainty

For any real number $r \in \mathbb{R}$ and any element $x$ in a unital algebra $A$, the scalar multiple of the identity element $r \cdot 1$ commutes with $x$. That is, $$(r \cdot 1) x = x (r \cdot 1)$$

Let $A$ be a star-algebra and $\omega$ be a state (a unital positive linear map) on $A$. For any observables $a, b \in \text{Observable } A$, let the centered observables be defined as $\tilde{a} = a - \omega(a)1$ and $\tilde{b} = b - \omega(b)1$, representing their fluctuations around their mean values. Then the Lie bracket of the centered observables is equal to the Lie bracket of the original observables: $$\llbracket a - \omega(a)1, b - \omega(b)1 \rrbracket = \llbracket a, b \rrbracket$$ where the bracket is defined as $\llbracket x, y \rrbracket = -\frac{i}{2}(xy - yx)$. This equality holds because centering an observable involves subtracting a scalar multiple of the identity, which commutes with all elements and thus does not contribute to the commutator.

Let $A$ be a $*$-algebra and $\omega: A \to \mathbb{C}$ be a state (a unital positive linear map). For any observables $a, b \in \text{Observable } A$, let $\delta a = a - \omega\langle a \rangle 1$ and $\delta b = b - \omega\langle b \rangle 1$ denote the centered observables (fluctuations). Then the expectation of the product of these fluctuations decomposes into a real symmetric part (the covariance) and an imaginary antisymmetric part (the expectation of the commutator): $$\omega(\delta a \delta b) = \text{Cov}_\omega(a, b) + i \omega\langle \llbracket a, b \rrbracket \rangle$$ where $\text{Cov}_\omega(a, b) = \text{Re}(\omega(\delta a \delta b))$ is the covariance of $a$ and $b$, and $\llbracket a, b \rrbracket = -\frac{i}{2}(ab - ba)$ is the Lie bracket (self-adjoint commutator) of the observables.

Let $A$ be a unital $C^*$-algebra and $\omega: A \to \mathbb{C}$ be a state (a unital positive linear map). Let $\pi_\omega: A \to \mathcal{B}(H_\omega)$ be the GNS representation on the GNS Hilbert space $H_\omega$, and let $\Omega_\omega \in H_\omega$ be the GNS cyclic vector. For any elements $x, y \in A$, the inner product of the vectors $\pi_\omega(x) \Omega_\omega$ and $\pi_\omega(y) \Omega_\omega$ in $H_\omega$ is given by: $$\langle \pi_\omega(x) \Omega_\omega, \pi_\omega(y) \Omega_\omega \rangle = \omega(x^* y)$$ where $x^*$ denotes the adjoint of $x$.

Let $A$ be a $*$-algebra and let $\omega$ be a state (represented as a unital positive linear map $\mathcal{S}[\mathbb{C}, A]$). For any elements $x, y \in A$, the Cauchy–Schwarz inequality holds: $$\|\omega(x^* y)\| \cdot \|\omega(y^* x)\| \le \text{Re}(\omega(x^* x)) \cdot \text{Re}(\omega(y^* y))$$ Given that for a state $\omega(y^* x) = \overline{\omega(x^* y)}$ and $\omega(x^* x)$ is a non-negative real number, this is equivalent to the standard form: $$|\omega(x^* y)|^2 \le \omega(x^* x) \omega(y^* y)$$ where $x^*$ denotes the adjoint of $x$, $\|\cdot\|$ denotes the complex norm (modulus), and $\text{Re}$ denotes the real part.

Let $A$ be a complex $*$-algebra and let $\omega: A \to \mathbb{C}$ be a state (represented as a unital positive linear map). For any two observables (self-adjoint elements) $a, b \in A$, let $\Delta a = a - \omega(a) \cdot 1$ and $\Delta b = b - \omega(b) \cdot 1$ denote the centered observables, representing the fluctuations of $a$ and $b$ around their mean values. Then the product of the norms of the correlations between these fluctuations satisfies: $$\|\omega(\Delta a \Delta b)\| \cdot \|\omega(\Delta b \Delta a)\| \le \text{Var}_\omega(a) \cdot \text{Var}_\omega(b)$$ where $\text{Var}_\omega(x) = \omega(\Delta x^2)$ is the variance of the observable $x$ in the state $\omega$, and $\|\cdot\|$ denotes the complex norm.

Let $A$ be a complex star-algebra and $\omega$ be a state (a unital positive linear map from $A$ to $\mathbb{C}$). For any two observables $a, b \in A$ (self-adjoint elements), let $\Delta a = a - \omega(a)1$ and $\Delta b = b - \omega(b)1$ be the centered observables representing the fluctuations around their mean values. Then the squared magnitude of the expectation of the product of these centered observables is bounded by the product of their variances: $$|\omega(\Delta a \Delta b)|^2 \le \text{Var}_\omega(a) \cdot \text{Var}_\omega(b)$$ where $\text{Var}_\omega(x) = \omega(\Delta x^2)$ is the variance of the observable $x$ in the state $\omega$, and $|\cdot|^2$ denotes the squared complex modulus.

Let $A$ be a complex star-algebra and $\omega: A \to \mathbb{C}$ be a state (a unital positive linear map). For any two observables $a, b \in \text{Observable } A$ (self-adjoint elements), the following inequality holds: $$\text{Cov}_\omega(a, b)^2 + \omega\langle \llbracket a, b \rrbracket \rangle^2 \le \text{Var}_\omega(a) \cdot \text{Var}_\omega(b)$$ where $\text{Var}_\omega$ denotes the variance, $\text{Cov}_\omega$ denotes the covariance, $\omega\langle \cdot \rangle$ denotes the expectation value, and $\llbracket a, b \rrbracket = -\frac{i}{2}(ab - ba)$ is the Lie bracket of the observables.

Let $A$ be a complex star-algebra and $\omega: A \to \mathbb{C}$ be a state (a unital positive linear map). For any two observables $a, b \in A$ (defined as self-adjoint elements where $a^* = a$), the square of their covariance $\text{Cov}_\omega(a, b)$ is bounded by the product of their variances $\text{Var}_\omega(a)$ and $\text{Var}_\omega(b)$: $$\text{Cov}_\omega(a, b)^2 \leq \text{Var}_\omega(a) \cdot \text{Var}_\omega(b)$$ Here, the covariance is defined as $\text{Cov}_\omega(a, b) = \text{Re}(\omega((a - \langle a \rangle)(b - \langle b \rangle)))$ and the variance is $\text{Var}_\omega(a) = \text{Cov}_\omega(a, a)$, representing the uncertainty of the observables in the state $\omega$.

Let $A$ be a complex star-algebra and $\omega: A \to \mathbb{C}$ be a state (a unital positive linear map). For any two observables (self-adjoint elements) $a, b \in A$, the square of the expectation value of their Lie bracket $\llbracket a, b \rrbracket$ is bounded by the product of their variances: $$\omega\langle \llbracket a, b \rrbracket \rangle^2 \le \text{Var}_\omega(a) \cdot \text{Var}_\omega(b)$$ where the Lie bracket is defined as $\llbracket a, b \rrbracket = -\frac{i}{2}(ab - ba)$ and $\text{Var}_\omega(a)$ is the variance of the observable $a$ in state $\omega$.

Let $A$ be a complex $*$-algebra and $\omega: A \to \mathbb{C}$ be a state (a unital positive linear map). For any two observables $a, b \in \text{Observable } A$, the centered Gram defect is the real number defined as the difference between the product of their variances and the squared norm of their centered pairing: $$\text{Var}_\omega(a) \cdot \text{Var}_\omega(b) - |\omega(\Delta a \Delta b)|^2$$ where $\Delta a = a - \omega\langle a \rangle 1$ and $\Delta b = b - \omega\langle b \rangle 1$ are the centered observables. This quantity represents the "slack" in the Robertson–Schrödinger uncertainty relation, corresponding to the remainder in the Cauchy–Schwarz inequality applied to the centered observables.

Let $A$ be a complex $*$-algebra and $\omega: A \to \mathbb{C}$ be a state (a unital positive linear map). For any two observables $a, b \in A$ (self-adjoint elements), the centered Gram defect is non-negative: $$0 \le \text{Var}_\omega(a) \cdot \text{Var}_\omega(b) - |\omega(\Delta a \Delta b)|^2$$ where $\Delta a = a - \omega\langle a \rangle 1$ and $\Delta b = b - \omega\langle b \rangle 1$ are the centered observables, and $\text{Var}_\omega$ denotes the variance in state $\omega$. This inequality follows from the positivity of the state applied to the centered observables, representing the Cauchy–Schwarz inequality for centered variables.

Let $A$ be a complex $*$-algebra and $\omega: A \to \mathbb{C}$ be a state (a unital positive linear map). For any two observables $a, b \in A$ (self-adjoint elements), let $\Delta a = a - \omega\langle a \rangle 1$ and $\Delta b = b - \omega\langle b \rangle 1$ denote the centered observables. The squared absolute value of the expectation of their product (the centered pairing) decomposes into the sum of the squared covariance and the squared expectation of their Lie bracket: $$|\omega(\Delta a \Delta b)|^2 = \text{Cov}_\omega(a, b)^2 + \omega\langle \llbracket a, b \rrbracket \rangle^2$$ where $\text{Cov}_\omega(a, b) = \text{Re}(\omega(\Delta a \Delta b))$ is the covariance and $\llbracket a, b \rrbracket = -\frac{i}{2}(ab - ba)$ is the Lie bracket of the observables.

Let $A$ be a complex $*$-algebra and $\omega: A \to \mathbb{C}$ be a state (a unital positive linear map). For any two observables $a, b \in \text{Observable } A$, the difference between the product of their variances and the squared expectation of their Lie bracket $\llbracket a, b \rrbracket$ is equal to the sum of the centered Gram defect and the squared covariance: $$\text{Var}_\omega(a) \cdot \text{Var}_\omega(b) - \omega\langle \llbracket a, b \rrbracket \rangle^2 = \text{Defect}_\omega(a, b) + \text{Cov}_\omega(a, b)^2$$ where: - $\text{Var}_\omega(x)$ is the variance of observable $x$ in state $\omega$. - $\text{Cov}_\omega(a, b)$ is the covariance of $a$ and $b$ in state $\omega$. - $\llbracket a, b \rrbracket = -\frac{i}{2}(ab - ba)$ is the Lie bracket of the observables. - $\text{Defect}_\omega(a, b) = \text{Var}_\omega(a) \text{Var}_\omega(b) - |\omega(\Delta a \Delta b)|^2$ is the centered Gram defect, representing the Cauchy–Schwarz defect for the centered observables $\Delta a = a - \omega\langle a \rangle 1$ and $\Delta b = b - \omega\langle b \rangle 1$. This identity shows that the "slack" or gap in the Robertson uncertainty relation $|\omega\langle \llbracket a, b \rrbracket \rangle| \leq \sigma_a \sigma_b$ is exactly the sum of the Cauchy–Schwarz defect and the squared covariance.

Let $A$ be a complex $*$-algebra and $\omega: A \to \mathbb{C}$ be a state (a unital positive linear map). For any two observables $a, b \in \text{Observable } A$, let $\text{Var}_\omega(a)$ and $\text{Var}_\omega(b)$ denote their variances, $\text{Cov}_\omega(a, b)$ their covariance, and $\llbracket a, b \rrbracket = -\frac{i}{2}(ab - ba)$ their Lie bracket. The Robertson–Schrödinger uncertainty relation is saturated, i.e., $$\text{Cov}_\omega(a, b)^2 + \omega\langle \llbracket a, b \rrbracket \rangle^2 = \text{Var}_\omega(a) \cdot \text{Var}_\omega(b)$$ if and only if the centered Gram defect of $a$ and $b$ in state $\omega$ is zero.

Let $A$ be a complex star-algebra and $\omega: A \to \mathbb{C}$ be a state (a unital positive linear map). For any two observables $a, b$ (self-adjoint elements of $A$), let $\text{Var}_\omega(a)$ and $\text{Var}_\omega(b)$ denote their variances, $\text{Cov}_\omega(a, b)$ their covariance, and $\llbracket a, b \rrbracket = -\frac{i}{2}(ab - ba)$ their Lie bracket. The Robertson uncertainty relation saturates, meaning the product of the variances equals the square of the expectation of the bracket: $$\text{Var}_\omega(a) \cdot \text{Var}_\omega(b) = \omega\langle \llbracket a, b \rrbracket \rangle^2$$ if and only if the centered Gram defect is zero (meaning the Cauchy–Schwarz inequality for the centered observables saturates) and the covariance between $a$ and $b$ is zero: $$\text{centeredGramDefect}_\omega(a, b) = 0 \quad \text{and} \quad \text{Cov}_\omega(a, b) = 0.$$ Here, the centered Gram defect is defined as $\text{Var}_\omega(a) \text{Var}_\omega(b) - |\omega((a - \omega\langle a \rangle)(b - \omega\langle b \rangle))|^2$.

Let $A$ be a complex star-algebra and $\omega: A \to \mathbb{C}$ be a state (a unital positive linear map). For any two observables $a, b$ (self-adjoint elements of $A$), let $\llbracket a, b \rrbracket = -\frac{i}{2}(ab - ba)$ denote the Lie bracket and $\omega\langle \cdot \rangle$ denote the expectation value. If the squared expectation value of the bracket satisfies $\omega\langle \llbracket a, b \rrbracket \rangle^2 = \frac{1}{4}$, then the product of their variances satisfies the bound: $$1 \le 4 \cdot \text{Var}_\omega(a) \cdot \text{Var}_\omega(b)$$ where the variance is defined as $\text{Var}_\omega(a) = \omega\langle (a - \omega\langle a \rangle)^2 \rangle$.

Let $\omega$ be a state (a unital positive linear map from a complex star-algebra $A$ to $\mathbb{C}$) and let $a$ and $b$ be observables (self-adjoint elements of $A$). If the expectation value of the Lie bracket $\llbracket a, b \rrbracket = -\frac{i}{2}(ab - ba)$ satisfies the normalization condition $$\omega\langle \llbracket a, b \rrbracket \rangle^2 = \frac{1}{4},$$ then the variances of $a$ and $b$ in the state $\omega$ are both strictly positive, i.e., $\text{Var}_\omega(a) > 0$ and $\text{Var}_\omega(b) > 0$.

For a positive definite complex $n \times n$ matrix $A$, its normalizer is the matrix $T$ defined such that $T A T^\dagger = I$, where $I$ is the identity matrix and $T^\dagger$ denotes the conjugate transpose. The matrix $T$ is constructed as the product $D^{-1/2} U^\dagger$, where $U$ is a unitary matrix whose columns are the eigenvectors of $A$ and $D^{-1/2}$ is the diagonal matrix consisting of the inverse square roots of the corresponding eigenvalues.

Let $A$ be an $n \times n$ complex positive definite matrix. Let $T$ be the normalizer matrix associated with $A$. Then the product of $T$, $A$, and the conjugate transpose of $T$ is the identity matrix: $$T A T^\dagger = I$$ where $T^\dagger$ denotes the conjugate transpose of $T$ and $I$ is the $n \times n$ identity matrix.

Let $K$ be an $n \times n$ complex Hermitian matrix. If the matrices $I - K$ and $I + K$ are positive semidefinite, where $I$ is the identity matrix, then the absolute value of every eigenvalue $\lambda_i$ of $K$ is at most 1, i.e., $|\lambda_i| \le 1$ for all $i$.

Let $A$ and $B$ be $n \times n$ complex matrices. If $A$ is positive definite, $B$ is Hermitian, and the matrices $A - B$ and $A + B$ are both positive semidefinite, then the norm of the determinant of $B$ is less than or equal to the real part of the determinant of $A$: $$\|\det B\| \leq \text{Re}(\det A)$$ (Note: Since $A$ is positive definite, its eigenvalues are positive real numbers, meaning $\det A$ is real and positive, thus $\text{Re}(\det A) = \det A$.)

For any $n \times n$ matrix $S$ with real entries and any real number $\epsilon$, the complexification of the matrix $S + \epsilon I$ (where $I$ is the identity matrix) is equal to the complexification of $S$ plus the complex scalar $\epsilon$ times the identity matrix. That is, $(S + \epsilon I)_{\mathbb{C}} = S_{\mathbb{C}} + \epsilon I$.

Let $S$ and $W$ be $n \times n$ real matrices. If the complex matrices $S - iW$ and $S + iW$ are positive semidefinite, then the absolute value of the determinant of $W$ is bounded by the determinant of $S$: $$|\det W| \le \det S$$

Given a state $\omega$ (a unital positive linear map) on a $*$-algebra $A$ and an indexed family of observables $\{a_j\}_{j \in \iota}$ (self-adjoint elements of $A$), the covariance matrix is the $\iota \times \iota$ real matrix $\Sigma$ whose entries are the covariances of pairs of observables. Specifically, the $(j, k)$-th entry is given by: $$\Sigma_{jk} = \text{Cov}_\omega(a_j, a_k)$$ where $\text{Cov}_\omega(a_j, a_k) = \text{Re}(\omega((a_j - \omega\langle a_j \rangle)(a_k - \omega\langle a_k \rangle)))$ is the covariance of observables $a_j$ and $a_k$ in the state $\omega$, and $\omega\langle \cdot \rangle$ denotes the expectation value.

Given a state $\omega$ (a unital positive linear map) on a star-algebra $A$ and a family of observables $\{a_j\}_{j \in \iota}$ (self-adjoint elements of $A$), the bracket matrix is the $\iota \times \iota$ real matrix whose entries are the expectation values of the Lie brackets of the observables. Specifically, the $(j, k)$-th entry of the matrix is given by: $$\Omega_{jk} = \omega\langle \llbracket a_j, a_k \rrbracket \rangle$$ where $\llbracket a_j, a_k \rrbracket = -\frac{i}{2}(a_j a_k - a_k a_j)$ is the Lie bracket of the observables and $\omega\langle \cdot \rangle$ denotes the expectation value.

Given a state $\omega$ (a unital positive linear map) on a unital $\star$-algebra $A$ and an observable $a$ (a self-adjoint element of $A$), the **centered GNS vector** is the element of the GNS Hilbert space $H_\omega$ corresponding to the centered observable $a - \omega(a) \cdot 1$. It is defined as: $$\pi_\omega(a - \omega(a) \cdot 1) \Omega_\omega$$ where $\omega(a)$ is the expectation value of $a$ in the state $\omega$, $\pi_\omega$ is the GNS representation, and $\Omega_\omega$ is the GNS cyclic vector. This vector represents the fluctuation of the observable $a$ around its mean within the GNS Hilbert space.

Let $\omega$ be a state (a unital positive linear map) on a unital $\star$-algebra $A$, and let $\{a_j\}_{j \in \iota}$ be a collection of observables (self-adjoint elements) in $A$. Let $\psi_j$ be the centered GNS vector in the Hilbert space $H_\omega$ corresponding to the observable $a_j$, defined as $\psi_j = \pi_\omega(a_j - \omega\langle a_j \rangle \mathbf{1}) \Omega_\omega$. The Gram matrix $G$ of these vectors, whose entries are the inner products $G_{jk} = \langle \psi_j, \psi_k \rangle_{H_\omega}$, is given by the sum of the covariance matrix $\Sigma$ and the imaginary bracket matrix $i\Omega$: $$G = \Sigma + i\Omega$$ where the entries of the covariance matrix are $\Sigma_{jk} = \text{Cov}_\omega(a_j, a_k)$ and the entries of the bracket matrix are $\Omega_{jk} = \omega\langle \llbracket a_j, a_k \rrbracket \rangle$, with $\llbracket a_j, a_k \rrbracket = -\frac{i}{2}(a_j a_k - a_k a_j)$ denoting the Lie bracket of the observables.

Let $\omega$ be a state (a unital positive linear map) on a unital $\star$-algebra $A$, and let $\{a_j\}_{j \in \iota}$ be a family of observables (self-adjoint elements) in $A$. Let $v_j = \pi_\omega(a_j - \omega\langle a_j \rangle 1) \Omega_\omega$ be the centered GNS vectors in the Hilbert space $H_\omega$. The transpose of the Gram matrix $G$ of these vectors, where $G_{jk} = \langle v_j, v_k \rangle$, is given by: $$G^\mathsf{T} = \Sigma - i\Omega$$ where: - $\Sigma$ is the covariance matrix with entries $\Sigma_{jk} = \text{Cov}_\omega(a_j, a_k) = \text{Re}(\omega((a_j - \omega\langle a_j \rangle)(a_k - \omega\langle a_k \rangle)))$, - $\Omega$ is the bracket matrix with entries $\Omega_{jk} = \omega\langle \llbracket a_j, a_k \rrbracket \rangle$, where $\llbracket a_j, a_k \rrbracket = -\frac{i}{2}(a_j a_k - a_k a_j)$ is the Lie bracket of the observables, - $i$ is the imaginary unit.

Let $\omega$ be a state (a unital positive linear map) on a $*$-algebra $A$ and $\{a_j\}_{j \in \iota}$ be a finite family of observables (self-adjoint elements of $A$). Let $\Omega$ be the bracket matrix defined by the expectation values of the Lie brackets of the observables, $\Omega_{jk} = \omega\langle \llbracket a_j, a_k \rrbracket \rangle$, where $\llbracket a_j, a_k \rrbracket = -\frac{i}{2}(a_j a_k - a_k a_j)$. Let $\Sigma$ be the covariance matrix defined by $\Sigma_{jk} = \text{Cov}_\omega(a_j, a_k)$. Then the following uncertainty relation holds: $$|\det \Omega| \le \det \Sigma$$