PhyslibAlpha.AlgebraicFramework.CStarAlgebra.Uncertainty

Let $A$ be a unital $C^*$-algebra and $\omega$ be a state on $A$. For any observables $a, b \in A$, let the centered observables be defined by $\tilde{a} = a - \omega(a)\mathbf{1}$ and $\tilde{b} = b - \omega(b)\mathbf{1}$, where $\mathbf{1}$ is the identity element of $A$. Then the commutator of the centered observables is equal to the commutator of the original observables: $$\llbracket \tilde{a}, \tilde{b} \rrbracket = \llbracket a, b \rrbracket$$ where $\llbracket x, y \rrbracket = xy - yx$ denotes the Lie bracket (commutator).

Let $\omega$ be a state (a unital positive linear map) on a $C^*$-algebra $A$. For any two observables $a, b \in A$, let $\Delta a = a - \omega(a)1$ and $\Delta b = b - \omega(b)1$ denote their centered fluctuations. The expectation value of the product of these fluctuations decomposes into a real symmetric part (the covariance) and an imaginary antisymmetric part (half the expectation of the commutator): $$\omega(\Delta a \Delta b) = \text{Cov}_\omega(a, b) + \frac{1}{2} \omega([a, b])$$ where $\text{Cov}_\omega(a, b) = \frac{1}{2} \omega(\Delta a \Delta b + \Delta b \Delta a)$ is the covariance and $[a, b] = ab - ba$ is the commutator of $a$ and $b$.

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

Let $A$ be a $C^*$-algebra and $\omega$ be a state on $A$ (a unital positive linear map). For any elements $x, y \in A$, the following inequality holds: $$|\omega(x^* y)| \cdot |\omega(y^* x)| \leq \text{Re}(\omega(x^* x)) \cdot \text{Re}(\omega(y^* y))$$ where $x^*$ denotes the adjoint of $x$, $|\cdot|$ denotes the complex modulus, and $\text{Re}(\cdot)$ denotes the real part.

Let $\omega$ be a state (a unital positive linear map) on a $C^*$-algebra $A$. For any observables $a, b \in A$, let $\bar{a} = a - \omega(a)1$ and $\bar{b} = b - \omega(b)1$ denote the centered observables (fluctuations). Then the product of the norms of the expectations of their cross-fluctuations is bounded by the product of their variances: $$\|\omega(\bar{a} \bar{b})\| \cdot \|\omega(\bar{b} \bar{a})\| \leq \text{Var}_\omega(a) \cdot \text{Var}_\omega(b)$$ where $\text{Var}_\omega(x) = \omega(\bar{x}^2)$ is the variance of the observable $x$ in the state $\omega$.

Let $A$ be a unital $C^*$-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 $\bar{a} = a - \omega(a)1$ and $\bar{b} = b - \omega(b)1$ be the centered observables. The squared magnitude of the expectation value of the product of these centered observables is bounded by the product of their variances: $$|\omega(\bar{a}\bar{b})|^2 \le \text{Var}_\omega(a) \cdot \text{Var}_\omega(b)$$ where $\text{Var}_\omega(x) = \omega(\bar{x}^2)$ denotes the variance of the observable $x$ with respect to the state $\omega$.

Let $A$ be a $C^*$-algebra and $\omega$ be a state (a unital positive linear map) on $A$. For any two observables (self-adjoint elements) $a, b \in A$, the Robertson–Schrödinger uncertainty inequality holds: $$\text{Cov}_\omega(a, b)^2 + \left( \frac{1}{2i} \omega([a, b]) \right)^2 \le \text{Var}_\omega(a) \cdot \text{Var}_\omega(b)$$ where $\text{Var}_\omega(a)$ and $\text{Var}_\omega(b)$ are the variances of $a$ and $b$ with respect to the state $\omega$, $\text{Cov}_\omega(a, b)$ is their covariance, and $[a, b] = ab - ba$ is the commutator of $a$ and $b$.

Let $\omega$ be a state on a $C^*$-algebra $A$. For any two observables $a$ and $b$ in $A$, the square of their covariance with respect to $\omega$ is less than or equal to the product of their variances: $$\text{cov}_\omega(a, b)^2 \leq \text{var}_\omega(a) \cdot \text{var}_\omega(b)$$ where $\text{cov}_\omega(a, b)$ denotes the covariance of $a$ and $b$, and $\text{var}_\omega(x)$ denotes the variance of an observable $x$ under the state $\omega$.

Let $\omega$ be a state on a unital $C^*$-algebra $A$. For any two observables $a, b \in A$, the square of the expectation value of their commutator (multiplied by $1/2i$ to ensure it is real) is less than or equal to the product of their variances: $$\omega\left( \frac{1}{2i} [a, b] \right)^2 \leq \text{Var}_\omega(a) \cdot \text{Var}_\omega(b)$$ where $[a, b] = ab - ba$ is the commutator and $\text{Var}_\omega(x) = \omega(x^2) - \omega(x)^2$ is the variance of the observable $x$ in the state $\omega$. This result is a version of Heisenberg's uncertainty relation obtained by neglecting the covariance term in the Robertson–Schrödinger inequality.

Let $\omega$ be a state (a unital positive linear map) on a $C^*$-algebra $A$, and let $a, b$ be observables (self-adjoint elements) in $A$. The **centered Gram defect** is the real-valued difference between the product of the variances of $a$ and $b$ and the squared absolute value of the expectation of the product of their centered fluctuations: $$\text{Var}_\omega(a) \text{Var}_\omega(b) - |\omega(\Delta a \Delta b)|^2$$ where $\Delta a = a - \omega(a)\mathbf{1}$ and $\Delta b = b - \omega(b)\mathbf{1}$ are the centered observables, and $\text{Var}_\omega(x) = \omega(\Delta x^2)$ is the variance of $x$ in the state $\omega$. This quantity represents the non-negative gap in the Robertson–Schrödinger uncertainty relation.

Let $\omega$ be a state (a unital positive linear map) on a $C^*$-algebra $A$. For any two observables (self-adjoint elements) $a, b \in A$, the centered Gram defect is non-negative: $$0 \leq \text{Var}_\omega(a) \text{Var}_\omega(b) - |\omega(\Delta a \Delta b)|^2$$ where $\Delta a = a - \omega(a)\mathbf{1}$ and $\Delta b = b - \omega(b)\mathbf{1}$ are the centered observables, and $\text{Var}_\omega(x) = \omega(\Delta x^2)$ is the variance of $x$ in the state $\omega$.

Let $A$ be a star-algebra and $\omega$ be a state (a unital positive linear map) on $A$. For any two observables (self-adjoint elements) $a, b \in A$, let $\Delta a = a - \omega(a)\mathbf{1}$ and $\Delta b = b - \omega(b)\mathbf{1}$ denote the centered observables representing the fluctuations of $a$ and $b$ about their mean values. The squared absolute value of the expectation of the product $\Delta a \Delta b$ is equal to the sum of the square of the covariance of $a$ and $b$ and the square of the expectation of their Lie bracket: $$|\omega(\Delta a \Delta b)|^2 = \text{Cov}_\omega(a, b)^2 + \omega(\llbracket a, b \rrbracket)^2$$ where $\text{Cov}_\omega(a, b) = \text{Re}(\omega(\Delta a \Delta b))$ and $\llbracket a, b \rrbracket = -\frac{i}{2}(ab - ba)$.

Let $A$ be a star-algebra and $\omega$ be a state (a unital positive linear map) on $A$. For any two observables $a, b \in A$ (self-adjoint elements), the difference between the product of their variances and the square of the expectation of their Lie bracket $\llbracket a, b \rrbracket = -\frac{i}{2}(ab - ba)$ is equal to the sum of the centered Gram defect and the square of their covariance: $$\text{Var}_\omega(a) \text{Var}_\omega(b) - \omega\langle \llbracket a, b \rrbracket \rangle^2 = \text{centeredGramDefect}_\omega(a, b) + \text{Cov}_\omega(a, b)^2$$ In this expression, $\text{Var}_\omega(x)$ denotes the variance of observable $x$, $\text{Cov}_\omega(a, b)$ is the covariance of $a$ and $b$, and the centered Gram defect is defined as $\text{Var}_\omega(a) \text{Var}_\omega(b) - |\omega(\Delta a \Delta b)|^2$, where $\Delta x = x - \omega\langle x \rangle \mathbf{1}$ represents the fluctuation of $x$ about its mean.

Let $\omega$ be a state (a unital positive linear map) on a $C^*$-algebra $A$, and let $a$ and $b$ be observables (self-adjoint elements) in $A$. The Robertson–Schrödinger uncertainty relation is saturated, satisfying the equality $$\text{Cov}_\omega(a, b)^2 + \omega(\llbracket a, b \rrbracket)^2 = \text{Var}_\omega(a) \text{Var}_\omega(b),$$ if and only if the centered Gram defect of $a$ and $b$ in the state $\omega$ is zero. Here, $\text{Cov}_\omega(a, b)$ denotes the covariance, $\text{Var}_\omega(x)$ the variance, and $\llbracket a, b \rrbracket = -\frac{i}{2}(ab - ba)$ the Lie bracket of the observables.

Let $\omega$ be a state (a unital positive linear map) on a $C^*$-algebra $A$. For any two observables $a, b \in A$ (self-adjoint elements), let $\llbracket a, b \rrbracket = -\frac{i}{2}(ab - ba)$ be their Lie bracket, $\text{Var}_\omega(x)$ be the variance of an observable, and $\text{Cov}_\omega(a, b)$ be their covariance. The Robertson uncertainty relation saturates, meaning $$\omega\langle\llbracket a, b \rrbracket\rangle^2 = \text{Var}_\omega(a) \text{Var}_\omega(b),$$ if and only if the centered Gram defect is zero ($\text{Var}_\omega(a) \text{Var}_\omega(b) - |\omega(\Delta a \Delta b)|^2 = 0$) and the covariance between the observables is zero ($\text{Cov}_\omega(a, b) = 0$). This equivalence holds generally, including cases where the variance is zero, without the need for division.

Let $\omega$ be a state (a unital positive linear map) on a $C^*$-algebra $A$, and let $a$ and $b$ be observables (self-adjoint elements). If the expectation value of their Lie bracket $\llbracket a, b \rrbracket = -\frac{i}{2}(ab - ba)$ satisfies the normalization condition $$\omega(\llbracket a, b \rrbracket)^2 = \frac{1}{4},$$ then the product of their variances satisfies the inequality $$1 \le 4 \text{Var}_\omega(a) \text{Var}_\omega(b),$$ where $\text{Var}_\omega(x)$ denotes the variance of an observable $x$ in the state $\omega$.

Let $\omega$ be a state (a unital positive linear map) on a $C^*$-algebra $A$. For any two observables $a, b \in A$ (self-adjoint elements), let $\llbracket a, b \rrbracket = -\frac{i}{2}(ab - ba)$ be their Lie bracket and $\text{Var}_\omega(x)$ be the variance of an observable $x$ in the state $\omega$. If the expectation value of the Lie bracket satisfies $$\omega\langle \llbracket a, b \rrbracket \rangle^2 = \frac{1}{4},$$ then the variances of $a$ and $b$ are strictly positive: $$\text{Var}_\omega(a) > 0 \quad \text{and} \quad \text{Var}_\omega(b) > 0.$$