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Mean, second moment and variance of the scalar spectral law

Statement

Assume the Axiom of Choice and the Axiom of Countable Choice (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $H$ be a nonzero complex Hilbert space (Hilbert space) and let $A$ be a self-adjoint operator on $H$ with spectral projection-valued measure $E_{A}$ and scalar measures $E_{A, ψ} (B) = ⟨ E_{A} (B) ψ, ψ ⟩$ (Spectral theorem for unbounded self-adjoint operators (PVM form), Projection valued measure, Scalar and complex measures from a pvm). For a unit vector $ψ \in H$ write $μ_{ψ}^{A} : = μ_{T_{ψ, ψ}}^{A}$ for the scalar law of the pure state $T_{ψ, ψ}$ of Density operators and scalar spectral probability clause (b). Then:

(a) Second moment. $\int_{R} λ^{2} d μ_{ψ}^{A} (λ) = ∥ A ψ ∥^{2}$ if $ψ \in D (A)$ , and the integral equals $+ \infty$ if $ψ \notin D (A)$ ; thus the second moment of the scalar law is finite exactly on $D (A)$ .

(b) Mean. $\int_{R} ∣ λ ∣ d μ_{ψ}^{A} (λ) < \infty$ if and only if $ψ \in D (∣ A ∣^{1 / 2})$ ; in that case the mean $m (ψ) : = \int_{R} λ d μ_{ψ}^{A} (λ)$ is finite and real, and if $ψ \in D (A)$ then $m (ψ) = ⟨ A ψ, ψ ⟩$ .

(c) Variance. If $ψ \in D (A)$ and $m : = ⟨ A ψ, ψ ⟩$ , then $\int_{R} (λ - m)^{2} d μ_{ψ}^{A} (λ) = ∥ A ψ ∥^{2} - m^{2} = ∥ (A - m) ψ ∥^{2} \geq 0;$ if in addition $ψ \in D (A^{2})$ , that is $ψ \in D (A)$ and $A ψ \in D (A)$ , then $∥ (A - m) ψ ∥^{2} = ⟨ A^{2} ψ, ψ ⟩ - ⟨ A ψ, ψ ⟩^{2}$ .

(d) States. Let $ρ$ be a density operator with expansion $ρ = \sum_{n} w_{n} T_{u_{n}, u_{n}}$ , $w_{n} \geq 0$ , $\sum_{n} w_{n} = 1$ , $(u_{n})$ orthonormal. In every weighted sum below, a term with $w_{n} = 0$ is defined to be zero before any component integral is evaluated; equivalently, sum only over $n$ with $w_{n} > 0$ . This convention also applies to signed component means, which need not exist at zero-weight indices. Then for every Borel function $g \geq 0$ $\int_{R} g d μ_{ρ}^{A} = \sum_{n} w_{n} \int_{R} g d E_{A, u_{n}} \in [0, \infty],$ so the mean of $μ_{ρ}^{A}$ is finite if and only if $\sum_{n} w_{n} \int ∣ λ ∣ d E_{A, u_{n}} < \infty$ , and then it equals $\sum_{n} w_{n} \int λ d E_{A, u_{n}}$ . For the second moment, put $M_{n} : = \int λ^{2} d E_{A, u_{n}}$ , so $M_{n} = ∥ A u_{n} ∥^{2}$ when $u_{n} \in D (A)$ and $M_{n} = + \infty$ otherwise; a zero weight contributes zero. Then $\int λ^{2} d μ_{ρ}^{A} = \sum_{n} w_{n} M_{n} \in [0, \infty]$ . It is finite exactly when every $u_{n}$ with $w_{n} > 0$ lies in $D (A)$ and $\sum_{n : w_{n} > 0} w_{n} ∥ A u_{n} ∥^{2} < \infty$ ; componentwise domain membership alone does not ensure a finite second moment.

Facts & Assumptions

Given: The Axiom of Choice and the Axiom of Countable Choice (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )); a nonzero complex Hilbert space $H$ ; a self-adjoint operator $A$ with PVM $E_{A}$ ; a unit vector $ψ \in H$ ; a density operator $ρ$ with expansion $ρ = \sum_{n} w_{n} T_{u_{n}, u_{n}}$ .

[F1]

Spectral domain and calculus. $D (A) = {ψ \in H : \int λ^{2} d E_{A, ψ} < \infty}$ and $A ψ = \int λ d E_{A} (λ) ψ$ on $D (A)$ ; for every Borel $f$ the operator $f (A)$ has domain $D (f (A)) = {ψ : \int ∣ f ∣^{2} d E_{A, ψ} < \infty}$ and satisfies $∥ f (A) ψ ∥^{2} = \int ∣ f ∣^{2} d E_{A, ψ}$ (Spectral theorem for unbounded self-adjoint operators (PVM form), Unbounded Borel functional calculus: domains, products, spectral mapping, Integral of a measurable function against a projection-valued measure).

[F2]

Mean and second moment of a pure state. For a unit vector $ψ \in D (A)$ one has $\int λ d E_{A, ψ} = ⟨ A ψ, ψ ⟩$ and $\int λ^{2} d E_{A, ψ} = ∥ A ψ ∥^{2}$ ; moreover $\int ∣ λ ∣ d E_{A, ψ} < \infty$ exactly when $ψ \in D (∣ A ∣^{1 / 2})$ , and then the mean $\int λ d E_{A, ψ}$ is finite (Density operators and scalar spectral probability clause (c)).

[F3]

Scalar law and its series form. For the density operator $ρ$ with the displayed expansion and every PVM $E$ , $tr (ρ E (B)) = \sum_{n} w_{n} ∥ E (B) u_{n} ∥^{2}$ for every Borel $B$ , and for a unit vector $u$ the law of $T_{u, u}$ is $∥ E (B) u ∥^{2} = E_{u} (B)$ (Density operators and scalar spectral probability clauses (a) and (b)).

[F4]

Monotone convergence and approximation. If $0 \leq f_{m} ↑ f$ pointwise then $\int f_{m} d ν ↑ \int f d ν$ for every measure $ν$ (Monotone convergence for the integral); every nonnegative measurable $g$ is the pointwise increasing limit of nonnegative simple functions ; and if $f_{m} \to f$ pointwise with $∣ f_{m} ∣ \leq G$ for an integrable $G$ then $\int f_{m} d ν \to \int f d ν$ (Dominated convergence).

[F5]

Self-adjoint quadratic forms are real. For a self-adjoint $A$ and $ψ \in D (A)$ one has $⟨ A ψ, ψ ⟩ = \overline{⟨ ψ, A ψ ⟩} = \overline{⟨ A ψ, ψ ⟩} \in R$ , and $∥ (A - m) ψ ∥^{2} = ∥ A ψ ∥^{2} - 2 m ⟨ A ψ, ψ ⟩ + m^{2} ∥ ψ ∥^{2}$ for real $m$ (Symmetric, self-adjoint and essentially self-adjoint operators).

Proof

1.1givenF1F2F5algebra

[Second moment and mean for a state vector] Let $ψ \in H$ be a unit vector and identify $μ_{ψ}^{A}$ with the finite measure $E_{A, ψ}$ . If $ψ \in D (A)$ , [F1] and [F2] give $\int λ^{2} d μ_{ψ}^{A} = ∥ A ψ ∥^{2} < \infty$ and $\int λ d μ_{ψ}^{A} = ⟨ A ψ, ψ ⟩$ ; if $ψ \notin D (A)$ , then [F1] gives $\int λ^{2} d μ_{ψ}^{A} = \infty$ , so the second moment is infinite. In the first case [F5] makes $⟨ A ψ, ψ ⟩$ real. This proves (a) and the mean formula of (b).

1.2givenF3F4algebra

[Reduction of state moments to pure moments] By [F3], $μ_{ρ}^{A} (B) = \sum_{n} w_{n} E_{A, u_{n}} (B)$ for every Borel $B$ . Let $g \geq 0$ be Borel, and omit all zero-weight indices as stipulated in (d). For indicators $g = 1_{B}$ the identity $\int g d μ_{ρ}^{A} = \sum_{n} w_{n} \int g d E_{A, u_{n}}$ is exactly [F3]; it extends to nonnegative simple $g$ by linearity, and to all nonnegative Borel $g$ by writing $g$ as an increasing pointwise limit of nonnegative simple functions [F4] and applying monotone convergence twice, to the measure $μ_{ρ}^{A}$ and to the partial sums of the series. This proves the displayed identity of (d).

2.1givenstep 1.1F1F3algebra

[The first moment criterion] For any unit $ψ \in H$ , define $∣ A ∣^{1 / 2} : = f (A)$ with the Borel function $f (λ) = ∣ λ ∣^{1 / 2}$ . By [F1], $D (∣ A ∣^{1 / 2}) = {ψ \in H : \int ∣ λ ∣ d E_{A, ψ} < \infty}$ . By [F3], $E_{A, ψ} = μ_{ψ}^{A}$ , so this gives the claimed equivalence without assuming $ψ \in D (A)$ . When the absolute first moment is finite, the positive and negative parts of $λ$ have finite integrals, and their difference defines a finite real mean; otherwise the mean is not a finite Lebesgue integral. Together with step 1.1 this proves (b).

2.2givenstep 1.1F1F2F5algebra

[Variance of a state vector] Let $ψ \in D (A)$ be a unit vector and $m : = ⟨ A ψ, ψ ⟩$ , a real number by [F5]. Then $A ψ - m ψ \in H$ and by [F1] applied to $f (λ) = λ - m$ , $∥ (A - m) ψ ∥^{2} = \int (λ - m)^{2} d E_{A, ψ}$ . Expanding the square and using [F2] for the integrals of $λ^{2}$ , $λ$ and $1$ gives $\int (λ - m)^{2} d μ_{ψ}^{A} = \int λ^{2} d μ_{ψ}^{A} - 2 m \int λ d μ_{ψ}^{A} + m^{2} = ∥ A ψ ∥^{2} - 2 m^{2} + m^{2} = ∥ A ψ ∥^{2} - m^{2}$ .

3.1step 2.2F1F5algebra

[The second identity for the variance] If moreover $ψ \in D (A^{2})$ , then $A ψ \in D (A)$ and the inner product $⟨ A^{2} ψ, ψ ⟩$ is defined and equals $⟨ A (A ψ), ψ ⟩ = ⟨ A ψ, A ψ ⟩ = ∥ A ψ ∥^{2}$ because $A$ is self-adjoint; substituting this into step 2.2 gives $\int (λ - m)^{2} d μ_{ψ}^{A} = ⟨ A^{2} ψ, ψ ⟩ - m^{2} = ⟨ A^{2} ψ, ψ ⟩ - ⟨ A ψ, ψ ⟩^{2}$ , which together with step 2.2 proves (c).

4.1step 1.1step 1.2F2F4algebra∎

[Moments of a state] All sums in this step run over positive-weight indices, with the zero-term convention of (d). With $g = ∣ λ ∣$ in step 1.2, the mean of $μ_{ρ}^{A}$ is finite exactly when $\sum_{n} w_{n} \int ∣ λ ∣ d E_{A, u_{n}} < \infty$ ; in that case every positive-weight component has finite absolute first moment, and applying the signed version of the identity to the positive and negative parts and subtracting gives $\int λ d μ_{ρ}^{A} = \sum_{n} w_{n} \int λ d E_{A, u_{n}}$ , using dominated convergence [F4] for the tail sums. With $g = λ^{2}$ , step 1.2 and [F2] give $\int λ^{2} d μ_{ρ}^{A} = \sum_{n} w_{n} M_{n}$ , where $M_{n} = \int λ^{2} d E_{A, u_{n}}$ is $∥ A u_{n} ∥^{2}$ for $u_{n} \in D (A)$ and $+ \infty$ otherwise, and terms of zero weight contribute zero. Since this is a nonnegative series, it is finite exactly when every component of positive weight lies in $D (A)$ and the weighted sum of the squared norms is finite; having each positive-weight component in $D (A)$ by itself does not imply convergence of that series.

Mean, second moment and variance of the scalar spectral law

Statement

Facts & Assumptions

Proof

Depends on

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