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Mean, second moment and variance of an assigned observable in a preparation

Physical scope

Conditional statements, within the framework's observable and probability postulate, about the moments of an assigned sharp observable in a preparation. For a pure preparation represented by a unit vector ψ, the second moment is finite exactly when ψ lies in D(A), the mean is finite exactly when ψ lies in D(|A|^{1/2}), and for ψ in D(A) the mean is the quadratic form and the variance is the squared norm of the centred vector. For a density-operator expansion ρ=ΣₙwₙT_{uₙ,uₙ}, the mean is finite exactly when Σₙwₙ∫|λ|dE_{A,uₙ}<∞, and the second moment is finite exactly when every positive-weight uₙ lies in D(A) and Σ_{n:wₙ>0}wₙ||Auₙ||²<∞. A preparation need not have finite moments automatically. The item asserts no apparatus error, no disturbance and no empirical result.

Statement

Adopt the postulates P1 and P2 (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes) and assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Let $(A, u_{A})$ be an assigned sharp observable with outcome law $μ^{A}$ in the reading of Assigned sharp observables and the outcome law of the framework, and let $ρ$ be a preparation. Then:

(a) Pure preparations. Let $ρ = T_{ψ, ψ}$ be the pure preparation of a unit vector $ψ \in D (A)$ . Then the mean of the outcome law exists and equals $⟨ A ψ, ψ ⟩$ , and the second moment exists and equals $∥ A ψ ∥^{2}$ : $m (ψ) = \int_{R} λ d μ_{ψ}^{A} (λ) = ⟨ A ψ, ψ ⟩, \int_{R} λ^{2} d μ_{ψ}^{A} (λ) = ∥ A ψ ∥^{2},$ both read in the powers of the declared unit $u_{A}$ of the observable. If $ψ \in D (A^{2})$ , the variance is $\int_{R} (λ - m (ψ))^{2} d μ_{ψ}^{A} (λ) = ⟨ A^{2} ψ, ψ ⟩ - ⟨ A ψ, ψ ⟩^{2} .$

(b) Variance and its hypotheses. For every unit $ψ \in D (A)$ the variance exists and equals ${Var}_{ψ} (A) = ∥ A ψ ∥^{2} - ⟨ A ψ, ψ ⟩^{2} = ∥ (A - ⟨ A ψ, ψ ⟩ I) ψ ∥^{2} \geq 0,$ without any hypothesis beyond $ψ \in D (A)$ ; no appeal to a discrete eigenbasis is made.

(c) Non-existence is possible. The identities of (a) and (b) are conditional on membership of $ψ$ in the stated domains. If $ψ \notin D (∣ A ∣^{1 / 2})$ then $\int_{R} ∣ λ ∣ d μ_{ψ}^{A} (λ) = \infty$ , so the mean of the outcome law does not exist as a finite real number; the mean is finite exactly when $ψ \in D (∣ A ∣^{1 / 2})$ , and then $ψ$ need not lie in $D (A)$ . If $ψ \notin D (A)$ then the second moment $\int λ^{2} d μ_{ψ}^{A}$ is infinite. In particular every preparation of the model is a unit ray or a density operator, and no preparation is equipped with finite moments automatically.

(d) Mixtures. Let $ρ = \sum_{n} w_{n} T_{u_{n}, u_{n}}$ be a spectral expansion of a preparation, $w_{n} \geq 0$ , $\sum_{n} w_{n} = 1$ , $(u_{n})$ orthonormal. In every weighted moment sum, zero-weight terms are defined to be zero before evaluating a component integral; equivalently, only positive-weight indices are summed. This applies also to component means that are undefined at zero-weight indices. Then 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 $\int λ d μ_{ρ}^{A} = \sum_{n} w_{n} \int λ d E_{A, u_{n}}$ . For the second moment, put $M_{n} : = \int λ^{2} d E_{A, u_{n}}$ , so $M_{n} = ∥ A u_{n} ∥^{2}$ for $u_{n} \in D (A)$ and $M_{n} = + \infty$ otherwise; a zero weight contributes zero. Then $\int λ^{2} d μ_{ρ}^{A} = \sum_{n} w_{n} M_{n} \in [0, \infty]$ , and it is finite exactly when every $u_{n}$ of positive weight lies in $D (A)$ and $\sum_{n : w_{n} > 0} w_{n} ∥ A u_{n} ∥^{2} < \infty$ . Thus componentwise domain membership alone does not guarantee a finite mixed-state second moment. Whenever the mixture has a finite absolute first moment, the identity of clause (b) of The Born outcome law: probability measure, mixtures, atoms and spectral support gives its mean as the absolutely convergent weighted average of the component means.

(e) Status. Clauses (a)-(d) are conditional consequences of P1 and P2 together with the mathematical moment results cited below; they assert no apparatus resolution, no measurement error, no disturbance and no empirical value. Whether a concrete preparation has a finite mean or variance for a concrete assigned observable is a question about the pair $(ρ, A)$ and the tested identification of the observable, not a consequence of the framework's postulates alone.

Facts & Assumptions

Given: The adopted postulates P1 and P2 (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes); the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice); an assigned sharp observable $(A, u_{A})$ with spectral law $μ^{A}$ and declared unit $u_{A}$ in the reading of Assigned sharp observables and the outcome law of the framework; a unit vector $ψ$ ; a preparation $ρ$ with spectral expansion $ρ = \sum_{n} w_{n} T_{u_{n}, u_{n}}$ (The state space: density operators, pure and mixed states, and purity, Density operators and scalar spectral probability clause (a)).

[F1]

The outcome law. For an assigned observable the preparation-dependent law is $μ_{ρ}^{A} (B) = tr (ρ E_{A} (B))$ , equal to $∥ E_{A} (B) ψ ∥^{2}$ for a pure preparation of a unit vector $ψ$ , and $B \mapsto μ_{ρ}^{A} (B)$ is a probability measure on the Borel sets of $R$ (Assigned sharp observables and the outcome law of the framework clauses (b) and (c), The Born outcome law: probability measure, mixtures, atoms and spectral support clause (a)).

[F2]

Moments of the scalar law. For a unit vector $ψ$ : the second moment satisfies $\int λ^{2} d μ_{ψ}^{A} = ∥ A ψ ∥^{2}$ if $ψ \in D (A)$ and equals $+ \infty$ otherwise; the mean satisfies $\int ∣ λ ∣ d μ_{ψ}^{A} < \infty$ exactly when $ψ \in D (∣ A ∣^{1 / 2})$ , and is then finite and real with $\int λ d μ_{ψ}^{A} = ⟨ A ψ, ψ ⟩$ for $ψ \in D (A)$ ; for $ψ \in D (A)$ the variance is $∥ A ψ ∥^{2} - ⟨ A ψ, ψ ⟩^{2} = ∥ (A - ⟨ A ψ, ψ ⟩ I) ψ ∥^{2}$ , and for $ψ \in D (A^{2})$ it is $⟨ A^{2} ψ, ψ ⟩ - ⟨ A ψ, ψ ⟩^{2}$ (Mean, second moment and variance of the scalar spectral law clauses (a)-(c)).

[F3]

Mixtures. For preparations $ρ_{n}$ with weights $w_{n} \geq 0$ , $\sum_{n} w_{n} = 1$ , the mixture $ρ = \sum_{n} w_{n} ρ_{n}$ is a preparation and $μ_{ρ}^{A} (B) = \sum_{n} w_{n} μ_{ρ_{n}}^{A} (B)$ for every Borel $B$ (Affinity and trace-norm continuity of the scalar spectral law clause (b), clauses (b) and (d)).

[F4]

State moments of a mixture. All sums use clause (d)'s zero-weight convention. For a density operator with expansion $ρ = \sum_{n} w_{n} T_{u_{n}, u_{n}}$ , the mean of $μ_{ρ}^{A}$ is finite exactly when $\sum_{n} w_{n} \int ∣ λ ∣ d E_{A, u_{n}} < \infty$ and is then $\sum_{n} w_{n} \int λ d E_{A, u_{n}}$ , while $\int λ^{2} d μ_{ρ}^{A} = \sum_{n} w_{n} M_{n} \in [0, \infty]$ , where $M_{n} = \int λ^{2} d E_{A, u_{n}}$ equals $∥ A u_{n} ∥^{2}$ on $D (A)$ and $+ \infty$ otherwise (Mean, second moment and variance of the scalar spectral law clause (d)).

[F5]

Preparations and domains. A preparation is a unit ray of the assigned Hilbert space or a density operator on it; being a preparation imposes no membership in $D (A)$ or $D (A^{2})$ beyond what is stated for the assigned observable $A$ (The state space of the framework: preparations, pure rays and density operators clauses (a)-(c), State and composition postulate P1: Hilbert spaces, preparations and composite systems clauses (ii)-(iii)).

Proof

1.1givenF1F2algebra

[Clause (a): mean and second moment of a pure preparation] Let $ρ = T_{ψ, ψ}$ with $∥ ψ ∥ = 1$ . By [F1] the law of the preparation is $μ_{ψ}^{A} (B) = ∥ E_{A} (B) ψ ∥^{2}$ ; by [F2] its second moment is $∥ A ψ ∥^{2}$ if $ψ \in D (A)$ (and $+ \infty$ otherwise), and its mean is $⟨ A ψ, ψ ⟩$ when $ψ \in D (A)$ ; when $ψ \in D (A^{2})$ , [F2] gives the displayed variance value. This proves (a).

1.2givenF2F5algebra

[Clause (c): domains and non-existence] By [F2], $\int ∣ λ ∣ d μ_{ψ}^{A}$ is finite exactly for $ψ \in D (∣ A ∣^{1 / 2})$ , and then the mean is finite; if $ψ \notin D (∣ A ∣^{1 / 2})$ the integral of $∣ λ ∣$ diverges and no finite mean exists, while $\int λ^{2} d μ_{ψ}^{A} = \infty$ whenever $ψ \notin D (A)$ . Membership of $ψ$ in these domains is not implied by being a preparation: [F5] says only that preparations are unit rays or density operators.

2.1givenstep 1.1F2algebra

[Clause (b): variance under the single hypothesis $ψ \in D (A)$ ] Let $ψ \in D (A)$ be a unit vector. By [F2] the variance equals $∥ A ψ ∥^{2} - ⟨ A ψ, ψ ⟩^{2}$ and also $∥ (A - m) ψ ∥^{2}$ with $m = ⟨ A ψ, ψ ⟩$ , a real number because $A$ is self-adjoint; this squared norm is nonnegative, so the variance exists, is finite and is nonnegative, and no eigenbasis or additional domain hypothesis is used.

3.1givenstep 1.1step 2.1F2F3F4algebra

[Clause (d): mixtures] Let $ρ = \sum_{n} w_{n} T_{u_{n}, u_{n}}$ be the spectral expansion, omitting zero-weight terms before evaluating any component moment as stipulated in (d). By [F4] the mean of $μ_{ρ}^{A}$ is finite exactly when $\sum_{n} w_{n} \int ∣ λ ∣ d E_{A, u_{n}} < \infty$ and then has the displayed value. The same series identity gives $\int λ^{2} d μ_{ρ}^{A} = \sum_{n} w_{n} M_{n}$ , with zero-weight terms omitted and $M_{n} = \int λ^{2} d E_{A, u_{n}} = ∥ A u_{n} ∥^{2}$ on $D (A)$ and $+ \infty$ off it by [F2]. A nonnegative series is finite exactly when all its positive-weight terms are finite and their weighted sum converges, so membership of each positive-weight $u_{n}$ in $D (A)$ is necessary but not sufficient. The mixture identity for the law itself is [F3]; taking $g = ∣ λ ∣$ and $λ^{2}$ in [F4] and combining with [F2] yields the stated criteria. Under the finite absolute first-moment condition, the component absolute first moments have finite weighted sum, so summing positive and negative parts separately gives the absolutely convergent weighted-mean identity.

4.1step 1.1step 1.2step 2.1step 3.1∎

[Clause (e) and conclusion] Steps 1.1, 1.2, 2.1 and 3.1 prove (a)-(d) as conditional statements from P1, P2 and the cited mathematics; the reminders recorded in (e) are visible in the hypotheses of those steps, and no apparatus, error, disturbance or empirical claim enters the argument.

Mean, second moment and variance of an assigned observable in a preparation

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

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