PhyslibAlpha.QuantumMechanics.OperatorAlgebra.States.Uncertainty

Let $A$ be a unital complex $C^*$-algebra and $\omega$ be a state on $A$. For any two observables $a, b \in A$, let $\delta a = a - \omega(a)\mathbb{1}$ and $\delta b = b - \omega(b)\mathbb{1}$ be the centered observables representing the fluctuations of $a$ and $b$ around their mean values. Then the Lie bracket of the centered observables is equal to the Lie bracket of the original observables: \[ \unicode{x2045} \delta a, \delta b \unicode{x2046} = \unicode{x2045} a, b \unicode{x2046} \] where the bracket $\unicode{x2045} a, b \unicode{x2046}$ is defined as the imaginary part of the product $ab$, which is proportional to the commutator $[a, b]$. This identity reflects that the commutator only depends on the fluctuations of the operators and is invariant under a shift by their expectation values.

Let $A$ be a complex unital $C^*$-algebra and $\omega$ be a state on $A$. For any two observables $a, b \in A$ (self-adjoint elements), let $\delta a = a - \omega\langle a \rangle \mathbb{1}$ and $\delta b = b - \omega\langle b \rangle \mathbb{1}$ be the centered observables representing the fluctuations of $a$ and $b$ around their mean values. The expectation value of the product of these fluctuations decomposes into a real symmetric part (the covariance) and an imaginary antisymmetric part (the expectation of the bracket) as follows: \[ \omega(\delta a \delta b) = \text{cov}_\omega(a, b) + i \omega\langle \unicode{x2045} a, b \unicode{x2046} \rangle \] where $\text{cov}_\omega(a, b)$ is the covariance of $a$ and $b$ in state $\omega$, $i$ is the imaginary unit, and $\unicode{x2045} a, b \unicode{x2046}$ is the Lie bracket of the observables, defined as the imaginary part of their product $\frac{1}{2i}(ab - ba)$.

Let $A$ be an operator algebra and $\omega$ be a state on $A$. For any elements $x, y \in A$, the following inequality holds: $$\|\omega(x^* y)\| \cdot \|\omega(y^* x)\| \le \operatorname{Re}(\omega(x^* x)) \cdot \operatorname{Re}(\omega(y^* y))$$ where $x^*$ denotes the adjoint (or star) of $x$, $\omega(a)$ is the expectation value of an element $a$ under the state $\omega$, and $\|\cdot\|$ denotes the complex modulus. This is the Cauchy–Schwarz inequality for the positive sesquilinear form (often associated with the GNS construction) induced by the state $\omega$.

Let $A$ be a $C^*$-algebra. For any state $\omega$ on $A$ and any two observables $a, b \in A$ (self-adjoint elements such that $a = a^*$ and $b = b^*$), the value of the state on the product $ba$ is the complex conjugate of the value of the state on the product $ab$. That is, $$\omega(ba) = \overline{\omega(ab)}.$$ This identity allows for the reversal of the order of two observables within a state by applying complex conjugation.

Let $A$ be a $C^*$-algebra and $\omega$ be a state on $A$. For any two observables $a, b \in A$ (self-adjoint elements), let $\bar{a} = a - \omega(a)\mathbb{1}$ and $\bar{b} = b - \omega(b)\mathbb{1}$ denote the centered observables, representing the fluctuations of $a$ and $b$ around their expectation values. Then the product of the magnitudes of the correlation and its adjoint is bounded by the product of their variances: $$\|\omega(\bar{a}\bar{b})\| \cdot \|\omega(\bar{b}\bar{a})\| \le \text{var}_\omega(a) \cdot \text{var}_\omega(b)$$ where $\|\cdot\|$ denotes the complex modulus and $\text{var}_\omega(x) = \omega(\bar{x}^2)$ is the variance of the observable $x$ in the state $\omega$. This inequality serves as the foundation for various uncertainty relations, asserting that the correlation between fluctuations cannot exceed the product of their spreads.

Let $A$ be a unital complex $C^*$-algebra and $\omega$ be a state on $A$. For any observables $a, b \in A$ (self-adjoint elements), let $\Delta a = a - \omega(a)\mathbb{1}$ and $\Delta b = b - \omega(b)\mathbb{1}$ denote the centered observables, where $\omega(a)$ is the expectation value. Then the following inequality holds: $$|\omega(\Delta a \Delta b)|^2 \leq \text{var}_\omega(a) \cdot \text{var}_\omega(b)$$ where $\text{var}_\omega(a) = \omega((\Delta a)^2)$ and $\text{var}_\omega(b) = \omega((\Delta b)^2)$ are the variances of the observables $a$ and $b$ in the state $\omega$.

Let $A$ be a unital complex $C^*$-algebra and $\omega$ be a state on $A$. For any two observables $a, b \in A$ (self-adjoint elements), the Robertson–Schrödinger uncertainty inequality holds: \[ \text{cov}_\omega(a, b)^2 + \omega \langle \unicode{x2045}a, b\unicode{x2046} \rangle^2 \le \text{var}_\omega(a) \cdot \text{var}_\omega(b) \] where $\text{cov}_\omega(a, b)$ is the covariance of the observables in state $\omega$, $\text{var}_\omega$ is the variance, and $\unicode{x2045}a, b\unicode{x2046} = \text{Im}(ab) = \frac{1}{2i}(ab - ba)$ is the Lie bracket representing the imaginary part of their product. This inequality jointly bounds the covariance and the expectation of the commutator by the product of the individual variances.

Let $A$ be a unital complex $C^*$-algebra and $\omega$ be a state on $A$. For any two observables $a, b \in A$ (represented as self-adjoint elements), the square of their covariance $\text{cov}_\omega(a, b)$ is bounded by the product of their respective variances $\text{var}_\omega(a)$ and $\text{var}_\omega(b)$: $$\text{cov}_\omega(a, b)^2 \leq \text{var}_\omega(a) \cdot \text{var}_\omega(b)$$ This inequality implies that two observables cannot be more correlated than the product of their uncertainties (standard deviations) allows, which corresponds to the familiar relation $|\text{cov}_\omega(a, b)| \leq \sigma_a \sigma_b$ obtained by omitting the commutator term in the Robertson–Schrödinger uncertainty relation.

Let $A$ be a $C^*$-algebra, $\omega$ be a state on $A$, and $a, b \in A$ be observables (self-adjoint elements). Let $\unicode{x2045}a, b\unicode{x2046} = \text{Im}(ab) = \frac{1}{2i}(ab - ba)$ denote the Lie bracket of the observables. The Robertson uncertainty relation states that the product of the variances of $a$ and $b$ in the state $\omega$ is bounded below by the square of the expectation value of their bracket: \[ \omega\langle \unicode{x2045}a, b\unicode{x2046} \rangle^2 \leq \text{var}_\omega(a) \cdot \text{var}_\omega(b) \] where $\omega\langle \cdot \rangle$ denotes the expectation value and $\text{var}_\omega(\cdot)$ denotes the variance.