PhyslibAlpha.QuantumMechanics.OperatorAlgebra.States.Statistics

Given a state $\omega$ on a $C^*$-algebra $A$, the expectation value is an $\mathbb{R}$-linear map from the space of observables to the real numbers $\mathbb{R}$. For an observable $a$ (a self-adjoint element $a \in A$ such that $a = a^*$), the expectation value $\omega \langle a \rangle$ is defined as the real part of the value obtained by applying the positive linear map associated with $\omega$ to $a$. Physically, this represents the mean value or average of repeated measurements of the observable $a$ on a system prepared in state $\omega$.

Given a state $\omega$ and an observable $a$ in an operator algebra, the notation $\omega \langle a \rangle$ represents the expectation value of the observable $a$ with respect to the state $\omega$.

Let $A$ be a $C^*$-algebra, $\omega$ be a state on $A$, and $a$ be an observable (a self-adjoint element $a \in A$ such that $a = a^*$). The value of the complex-valued state functional $\omega$ applied to $a$ is equal to the real-valued expectation value $\omega \langle a \rangle$ of the observable $a$, when the latter is treated as a complex number.

Let $A$ be a unital $C^*$-algebra and $\omega$ be a state on $A$. The expectation value of the identity observable $1$ in the state $\omega$, denoted by $\omega\langle 1 \rangle$, is equal to $1$.

Let $A$ be a unital $C^*$-algebra. For any state $\omega$ on $A$ and any observable $a \in A$ (a self-adjoint element), if $a$ is a positive element (denoted $a \geq 0$), then the expectation value of $a$ in the state $\omega$, denoted by $\omega\langle a \rangle$, is non-negative: $\omega\langle a \rangle \geq 0$.

For a state $\omega$ on a $C^*$-algebra $A$ and an observable $a$, the centered observable is defined as $a - \omega\langle a \rangle \mathbb{1}$, where $\omega\langle a \rangle$ is the expectation value of $a$ in the state $\omega$ and $\mathbb{1}$ is the identity element of $A$. This represents the fluctuation of the observable $a$ around its mean value, ensuring that the expectation value of the resulting centered observable is zero.

Let $A$ be a unital $C^*$-algebra. For any state $\omega$ on $A$ and any observable $a \in A$, the expectation value of the centered observable (the fluctuation $a - \omega\langle a \rangle \mathbb{1}$) is zero: $$\omega\langle a - \omega\langle a \rangle \mathbb{1} \rangle = 0$$ where $\omega\langle a \rangle$ denotes the expectation value of $a$ in state $\omega$ and $\mathbb{1}$ is the identity element of $A$.

Let $\omega$ be a state on a $C^*$-algebra $A$ and $a \in A$ be an observable (a self-adjoint element). For any real number $c \in \mathbb{R}$, the centered observable of $a$ shifted by a constant $c \mathbb{1}$ is equal to the centered observable of $a$: $$\text{centered}(\omega, a + c\mathbb{1}) = \text{centered}(\omega, a)$$ where $\text{centered}(\omega, a) = a - \omega\langle a \rangle \mathbb{1}$ denotes the fluctuation of the observable $a$ around its expectation value $\omega\langle a \rangle$, and $\mathbb{1}$ is the identity element of the algebra.

For a state $\omega$ on a $C^*$-algebra $A$ and two observables $a, b \in A$, the covariance is the expectation value of the Jordan product of the centered observables: $$\text{cov}_\omega(a, b) = \omega \langle (a - \omega\langle a \rangle \mathbb{1}) \odot (b - \omega\langle b \rangle \mathbb{1}) \rangle$$ where $\omega\langle a \rangle$ is the expectation value of $a$, $\mathbb{1}$ is the identity element, and $x \odot y = \frac{1}{2}(xy + yx)$ is the Jordan product. This quantity represents the correlation between the fluctuations of the two observables in the state $\omega$.

For a state $\omega$ on a $C^*$-algebra $A$ and an observable $a \in A$, the variance $\text{var}_\omega(a)$ is defined as the covariance of the observable with itself: $$\text{var}_\omega(a) = \text{cov}_\omega(a, a) = \omega((a - \omega\langle a \rangle \mathbb{1})^2)$$ where $\omega\langle a \rangle$ is the expectation value of $a$ in the state $\omega$ and $\mathbb{1}$ is the identity element of the algebra. This quantity represents the spread of the measurement outcomes of $a$ about its mean, serving as the quantum analogue of the variance of a random variable.

Let $A$ be a unital $C^*$-algebra. For any state $\omega$ on $A$ and any observable $a \in A$, the covariance of $a$ with itself is equal to the variance of $a$: $$\text{cov}_\omega(a, a) = \text{var}_\omega(a)$$

Let $A$ be a unital complex $C^*$-algebra, $\omega$ be a state on $A$, and $a \in A$ be an observable (a self-adjoint element). The variance of $a$ in the state $\omega$, denoted $\text{var}_\omega(a)$, is equal to the real part of the expectation value of the square of the centered observable: $$\text{var}_\omega(a) = \text{Re}(\omega((a - \omega\langle a \rangle \mathbb{1})^2))$$ where $a - \omega\langle a \rangle \mathbb{1}$ is the centered observable representing the fluctuation of $a$ about its mean $\omega\langle a \rangle$, and $\mathbb{1}$ is the identity element of the algebra.

Let $A$ be a unital complex $C^*$-algebra. For any state $\omega$ on $A$ and any two observables $a, b \in A$, the covariance between $a$ and $b$ in the state $\omega$ is symmetric, satisfying $\text{cov}_\omega(a, b) = \text{cov}_\omega(b, a)$.

Let $A$ be a unital $C^*$-algebra and $\omega$ be a state on $A$. For any two observables $a, b \in A$ (self-adjoint elements) and any real number $c \in \mathbb{R}$, the covariance of the shifted observable $a + c\mathbb{1}$ and $b$ is equal to the covariance of $a$ and $b$: $$\text{cov}_\omega(a + c\mathbb{1}, b) = \text{cov}_\omega(a, b)$$ where $\mathbb{1}$ denotes the identity element of the algebra. This reflects the property that covariance depends only on the fluctuations of the observables, and is therefore invariant under translations by a constant.

For a state $\omega$ on a $C^*$-algebra $A$ and any two observables $a, b \in A$, shifting the second observable by a constant $c \in \mathbb{R}$ leaves the covariance unchanged: $$\text{cov}_\omega(a, b + c \mathbb{1}) = \text{cov}_\omega(a, b)$$ where $\mathbb{1}$ is the identity element of $A$. This property demonstrates that covariance depends only on the fluctuations of the observables, and adding a constant value to an observable does not affect its correlation with others.

Let $A$ be a unital complex $C^*$-algebra, $\omega$ be a state on $A$ (a positive linear functional with unit norm), and $a \in A$ be an observable (a self-adjoint element). The variance of $a$ in the state $\omega$, denoted as $\text{var}_\omega(a)$, is always non-negative: $$\text{var}_\omega(a) \ge 0$$ This result states that the spread of repeated measurements of an observable about its mean can never be negative, which ensures that variance is a physically valid measure of statistical uncertainty. Algebraically, this follows from the positivity of the state applied to the square of the centered observable $(a - \omega\langle a \rangle \mathbb{1})^2$.