PhyslibAlpha.AlgebraicFramework.CStarAlgebra.SpectralMeasure

Given a state $\omega$ and an observable $a$, the spectral functional is the positive linear map from the space of continuous real-valued functions on the spectrum of $a$, denoted $C(\sigma(a), \mathbb{R})$, to $\mathbb{R}$, defined by the assignment $f \mapsto \omega(f(a))$, where $f(a)$ is the result of the continuous functional calculus.

Given a state $\omega$ and an observable $a$, this definition provides the positive linear map from the space of compactly supported continuous real-valued functions on the spectrum of $a$, denoted $C_c(\sigma(a), \mathbb{R})$, to $\mathbb{R}$. It maps each function $f$ to $\omega(f(a))$, where $f(a)$ is the result of the continuous functional calculus applied to $a$. This specific construction on compactly supported functions is used to satisfy the requirements of the Riesz–Markov–Kakutani representation theorem to produce the spectral measure.

Given a state $\omega$ and an observable $a$, the spectral measure is the unique Borel measure $\mu_{\omega, a}$ on the spectrum $\sigma(a) \subseteq \mathbb{R}$ obtained via the Riesz–Markov–Kakutani representation theorem from the positive linear functional $f \mapsto \omega(f(a))$. For any continuous real-valued function $f$ on the spectrum, the integral of $f$ with respect to this measure satisfies $\int_{\sigma(a)} f \, d\mu_{\omega, a} = \omega(f(a))$, where $f(a)$ is defined by the continuous functional calculus.

For any state $\omega$ and observable $a$ in a $C^*$-algebra, the spectral measure $\mu_{\omega, a}$ defined on the spectrum $\sigma(a) \subseteq \mathbb{R}$ is a finite measure.

Let $\omega$ be a state on a $C^*$-algebra and $a$ be an observable. Let $\sigma(a) \subseteq \mathbb{R}$ denote the spectrum of $a$. The spectral functional associated with $\omega$ and $a$ is the positive linear map from the space of continuous real-valued functions on the spectrum $C(\sigma(a), \mathbb{R})$ to $\mathbb{R}$ defined by $f \mapsto \omega(f(a))$. This theorem states that evaluating this functional at the constant function $1$ yields the value $1$.

Let $\omega$ be a state and $a$ be an observable in a $C^*$-algebra $A$. Let $\sigma(a) \subseteq \mathbb{R}$ denote the spectrum of $a$. The spectral measure $\mu_{\omega, a}$ associated with $\omega$ and $a$, which is defined on $\sigma(a)$, is a probability measure (meaning its total mass is $1$, consistent with the property that $\omega(1) = 1$).

Let $\omega$ be a state on a $C^*$-algebra $A$ and $a$ be an observable in $A$ with spectrum $\sigma(a) \subseteq \mathbb{R}$. For any continuous real-valued function $f \in C(\sigma(a), \mathbb{R})$, the expectation value of the observable $f(a)$ (constructed via the continuous functional calculus) in the state $\omega$ is equal to the integral of $f$ with respect to the spectral measure $\mu_{\omega, a}$ on the spectrum: $$\omega(f(a)) = \int_{\sigma(a)} f(x) \, d\mu_{\omega, a}(x)$$

Given a state $\omega$ and an observable $a$, the real spectral measure is the probability measure $\mu_{\omega, a}$ on $\mathbb{R}$ obtained by pushing forward the spectral measure (initially defined on the spectrum $\sigma(a) \subseteq \mathbb{R}$) along the inclusion map $\iota: \sigma(a) \hookrightarrow \mathbb{R}$. This measure is characterized by the property that for any suitable function $f: \mathbb{R} \to \mathbb{R}$, its integral with respect to $\mu_{\omega, a}$ yields the expectation value $\omega(f(a))$.

Let $\omega$ be a state and $a$ be an observable in a $C^*$-algebra $A$. The real spectral measure $\mu_{\omega, a}$ on $\mathbb{R}$ associated with the preparation $\omega$ and observable $a$ is a probability measure.

Let $\omega$ be a state and $a$ be an observable. Let $\mu_{\omega, a}$ be the real spectral measure associated with $\omega$ and $a$, and let $\sigma(a) \subseteq \mathbb{R}$ denote the spectrum of $a$. Then the measure of the complement of the spectrum $\sigma(a)$ is zero: $$\mu_{\omega, a}(\mathbb{R} \setminus \sigma(a)) = 0$$ This signifies that the probability measure $\mu_{\omega, a}$ is concentrated on the spectrum of the observable $a$.

Let $\omega$ be a state and $a$ be an observable with spectrum $\sigma(a) \subseteq \mathbb{R}$. For any function $f: \mathbb{R} \to \mathbb{R}$ that is continuous on the spectrum $\sigma(a)$, the expectation value of the observable $f(a)$ (defined via the continuous functional calculus) in the state $\omega$ is given by the integral of $f$ with respect to the real spectral measure $\mu_{\omega, a}$: $$\omega(f(a)) = \int_{\mathbb{R}} f(y) \, d\mu_{\omega, a}(y)$$

Let $a$ be an observable and $\sigma(a) \subseteq \mathbb{R}$ be its spectrum. For any continuous function $g : \sigma(a) \to \mathbb{R}$ defined on the spectrum, there exists a function $f : \mathbb{R} \to \mathbb{R}$ that is continuous on $\sigma(a)$ and satisfies $f(x) = g(x)$ for all $x \in \sigma(a)$.

Let $\omega$ be a state and $a$ be an observable with spectrum $\sigma(a) \subseteq \mathbb{R}$. Let $\mu$ be a measure on $\mathbb{R}$ and $\iota : \sigma(a) \to \mathbb{R}$ be the inclusion map. Suppose the following conditions hold: 1. For every function $f: \mathbb{R} \to \mathbb{R}$ that is continuous on $\sigma(a)$, the expectation of the observable $f(a)$ (defined via continuous functional calculus) in state $\omega$ is given by the integral of $f$ against $\mu$: $$\omega(f(a)) = \int_{\mathbb{R}} f \, d\mu$$ 2. The measure $\mu$ is concentrated on the spectrum, such that the pushforward of its pullback along the inclusion $\iota$ recovers the original measure: $\iota_*(\iota^* \mu) = \mu$. Then, for any continuous function $g : \sigma(a) \to \mathbb{R}$, the integral of $g$ with respect to the pullback measure $\iota^* \mu$ is equal to the integral of $g$ with respect to the spectral measure $\mu_{\omega, a}$: $$\int_{\sigma(a)} g \, d(\iota^* \mu) = \int_{\sigma(a)} g \, d\mu_{\omega, a}$$

Let $\omega$ be a state and $a$ be an observable with spectrum $\sigma(a) \subseteq \mathbb{R}$. Suppose $\mu$ is a probability measure on $\mathbb{R}$ satisfying the following conditions: 1. $\mu$ is concentrated on the spectrum of $a$, i.e., $\mu(\mathbb{R} \setminus \sigma(a)) = 0$. 2. For every function $f: \mathbb{R} \to \mathbb{R}$ that is continuous on $\sigma(a)$, the integral of $f$ against $\mu$ recovers the expectation value of the observable $f(a)$ in the state $\omega$: $$\int_{\mathbb{R}} f \, d\mu = \omega(f(a))$$ Then $\mu$ is equal to the real spectral measure $\mu_{\omega, a}$.

Given an observable $a$ and a point $x$ in its real spectrum $\sigma(a)$ that is isolated (specifically, the singleton set $\{x\}$ is clopen in the subspace topology of the spectrum), `eigenIndicator` is the continuous function $\mathbb{1}_{\{x\}}: \sigma(a) \to \mathbb{R}$ which maps $y \in \sigma(a)$ to $1$ if $y = x$ and to $0$ otherwise.

For an observable $a$ and a point $x$ in its real spectrum $\sigma_{\mathbb{R}}(a)$ such that the singleton set $\{x\}$ is clopen (i.e., $x$ is an isolated point), the continuous indicator function $\mathbb{1}_{\{x\}}$ evaluated at any point $y \in \sigma_{\mathbb{R}}(a)$ is equal to the set-theoretic indicator function $\mathbf{1}_{\{x\}}(y)$, which takes the value $1$ if $y = x$ and $0$ otherwise.

Given an observable $a$ and a point $x$ in its real spectrum $\sigma(a)$ that is isolated (specifically, the singleton set $\{x\}$ is clopen in the subspace topology of the spectrum), the `eigenEffect` is the spectral projection $\mathbb{1}_{\{x\}}(a)$ obtained by applying the continuous functional calculus of $a$ to the indicator function $\mathbb{1}_{\{x\}}$. This operator is an effect—meaning it satisfies $0 \le \mathbb{1}_{\{x\}}(a) \le I$—and is idempotent, making it a genuine projection in the operator-algebraic sense that represents the measurement of whether $a$ takes the value $x$.

Let $a$ be an observable and let $x$ be a point in its real spectrum $\sigma(a)$ such that the singleton set $\{x\}$ is clopen in the subspace topology (i.e., $x$ is an isolated point of the spectrum). Then the continuous indicator function $\mathbb{1}_{\{x\}} \in C(\sigma(a), \mathbb{R})$, which maps $x$ to $1$ and all other points in the spectrum to $0$, is idempotent under multiplication: $$\mathbb{1}_{\{x\}} \cdot \mathbb{1}_{\{x\}} = \mathbb{1}_{\{x\}}$$

Let $a$ be an observable and let $x \in \sigma(a)$ be a point in its spectrum such that the singleton set $\{x\}$ is clopen in the subspace topology (i.e., $x$ is an isolated point of the spectrum). Then the spectral projection $\mathbb{1}_{\{x\}}(a)$ (the `eigenEffect`), obtained by applying the continuous functional calculus to the indicator function of $\{x\}$, is an idempotent element: $$\mathbb{1}_{\{x\}}(a)^2 = \mathbb{1}_{\{x\}}(a)$$

Let $a$ be an observable and let $x \in \sigma(a)$ be an isolated point in its real spectrum (such that the singleton set $\{x\}$ is clopen in the subspace topology of the spectrum). Then the corresponding spectral effect $\mathbb{1}_{\{x\}}(a)$ is sharp, meaning it is a projection.

Given an observable $a$ and an isolated point $x$ in its real spectrum $\sigma(a)$ (such that the singleton set $\{x\}$ is clopen in the subspace topology of the spectrum), the `eigenMeasurement` is a measurement with two possible outcomes, `true` and `false`. The effect associated with the outcome `true` is the spectral projection $\mathbb{1}_{\{x\}}(a)$ (the `eigenEffect`), and the effect associated with the outcome `false` is its complement $1 - \mathbb{1}_{\{x\}}(a)$. This measurement represents the binary question: "Does the observable $a$ result in the value $x$?"

For a state $\omega$ and an observable $a$, let $x$ be an isolated point in the real spectrum $\sigma(a)$ (specifically, such that the singleton set $\{x\}$ is clopen in the subspace topology of the spectrum). The probability of obtaining the `true` outcome in the binary measurement `eigenMeasurement` (which represents the question "does the measurement of $a$ result in $x$?") when the system is in state $\omega$ is equal to the value of the real spectral measure $\mu_{\omega, a}$ evaluated at the singleton set $\{x\}$.