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The Robertson relation for preparation spreads of assigned sharp observables

Physical scope

Conditional statements about the preparation spreads of two assigned sharp observables of the framework in a pure preparation of a unit vector: the weak Robertson bound Delta A Delta B >= |Im<A psi, B psi>| for unit vectors in both domains, the Robertson bound Delta A Delta B >= |<[A,B]psi,psi>|/2 and its anticommutator refinement when the product domains also contain the vector, and the automatic validity of these hypotheses for bounded observables. The statement is a conditional consequence of P1 and P2 and the cited mathematics; it concerns preparation spreads of outcome laws only, and asserts no apparatus error, no disturbance, no joint measurement, no energy-time inequality and no empirical result.

Statement

Adopt the state and composition postulate P1 and the observable and probability postulate P2 in the readings of The state space of the framework: preparations, pure rays and density operators and Assigned sharp observables and the outcome law of the framework (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 $(A, u_{A})$ and $(B, u_{B})$ be assigned sharp observables with outcome laws $μ^{A}, μ^{B}$ and let $Δ_{ρ} A, Δ_{ρ} B$ be their preparation spreads in the reading of Preparation variance and spread of an assigned sharp observable. Let $ψ$ be a unit vector. Whenever $ψ \in D (A) \cap D (B)$ , put $a : = ⟨ A ψ, ψ ⟩, b : = ⟨ B ψ, ψ ⟩ .$ Then:

(a) Weak form. If $ψ \in D (A) \cap D (B)$ , then the means $a, b$ are real, the spreads $Δ_{ψ} A$ and $Δ_{ψ} B$ are defined, both are read in the units $u_{A}$ and $u_{B}$ , and $Δ_{ψ} A Δ_{ψ} B \geq ∣ Im ⟨ A ψ, B ψ ⟩ ∣ .$

(b) Robertson form. If in addition $ψ \in D (A B) \cap D (B A)$ , that is $B ψ \in D (A)$ and $A ψ \in D (B)$ , then the pairing $⟨ [A, B] ψ, ψ ⟩ : = ⟨ A B ψ, ψ ⟩ - ⟨ B A ψ, ψ ⟩$ is defined, is purely imaginary, and $Δ_{ψ} A Δ_{ψ} B \geq \frac{1}{2} ∣ ⟨ [A, B] ψ, ψ ⟩ ∣,$ read in the product unit $u_{A} u_{B}$ ; moreover the anticommutator refinement $(Δ_{ψ} A)^{2} (Δ_{ψ} B)^{2} \geq (\frac{1}{2} ⟨ {(A - a), (B - b)} ψ, ψ ⟩)^{2} + \frac{1}{4} ∣ ⟨ [A, B] ψ, ψ ⟩ ∣^{2}$ holds, with equality in the refined bound if and only if $(A - a) ψ$ and $(B - b) ψ$ are linearly dependent.

(c) Bounded observables. If $A$ and $B$ are bounded everywhere-defined self-adjoint operators, then clauses (a) and (b) hold for every unit $ψ \in H$ with all hypotheses automatic (A bounded linear operator between normed spaces).

(d) Status and scope. Clauses (a)-(c) are conditional consequences of P1 and P2 and the cited mathematics. They concern the spreads of the two outcome laws in one preparation; they introduce no joint outcome law, no apparatus resolution, no measurement error, no disturbance of the preparation, and no time observable. In particular no energy-time inequality is asserted, and clauses (a)-(c) do not claim that a laboratory procedure realizes either observable or attains the bounds; whether a concrete preparation and apparatus realize the pair $(A, u_{A})$ and produce the spread $Δ_{ψ} A$ is a tested identification under the preparation and apparatus postulate P7.

Facts & Assumptions

Given: The postulates P1 and P2; the Axiom of Countable Choice and the Axiom of Choice; assigned sharp observables $(A, u_{A})$ , $(B, u_{B})$ with self-adjoint operators $A, B$ and declared units; a unit vector $ψ$ ; the preparation spreads of Preparation variance and spread of an assigned sharp observable.

[F1]

Moments and spread. For an assigned sharp observable $(A, u_{A})$ and a pure preparation of a unit vector $ψ \in D (A)$ , the mean and the second moment of the outcome law exist with $m = ⟨ A ψ, ψ ⟩ \in R$ and $\int λ^{2} d μ_{ψ}^{A} = ∥ A ψ ∥^{2}$ , and the preparation spread is $Δ_{ψ} A = ∥ (A - ⟨ A ψ, ψ ⟩ I) ψ ∥$ (Mean, second moment and variance of an assigned observable in a preparation clause (a), Mean, second moment and variance of the scalar spectral law clause (c), Preparation variance and spread of an assigned sharp observable clauses (b) and (c)).

[F2]

Schroedinger-Robertson bound. For symmetric densely defined $A, B$ and a unit $u \in D (A) \cap D (B)$ with $u \in D (A B) \cap D (B A)$ , with real means $a, b$ and centered vectors $x = (A - a) u$ , $y = (B - b) u$ , the refined bound and the equality clause hold: $(Δ A)^{2} (Δ B)^{2} \geq [Re ⟨ x, y ⟩]^{2} + [Im ⟨ x, y ⟩]^{2}$ , with equality if and only if $x, y$ are linearly dependent, and $Δ A Δ B \geq \frac{1}{2} ∣ ⟨ [A, B] u, u ⟩ ∣$ with $⟨ [A, B] u, u ⟩ = - 2 i Im ⟨ x, y ⟩$ ; and $Re ⟨ x, y ⟩ = \frac{1}{2} ⟨ {(A - a), (B - b)} u, u ⟩$ (The Schroedinger-Robertson bound in anticommutator form and the Robertson bound on product domains clauses (a)-(d)).

[F3]

Weak-form identification without products. For symmetric densely defined $A, B$ and a unit $u \in D (A) \cap D (B)$ , with real means $a, b$ and $x, y$ as above, $Im ⟨ x, y ⟩ = Im ⟨ A u, B u ⟩$ because $⟨ x, y ⟩ = ⟨ A u, B u ⟩ - a b$ with $a b \in R$ , and the refined Cauchy-Schwarz bound yields $Δ A Δ B \geq ∣ Im ⟨ A u, B u ⟩ ∣$ (The Schroedinger-Robertson bound in anticommutator form and the Robertson bound on product domains clause (a), Weak-form Robertson uncertainty on a common domain clauses (a), (c)).

[F4]

Units. The declared unit of an assigned sharp observable and the reading of powers and products of units are those of Dimensions, SI unit conventions and quantity types of the framework; the commutator pairing $⟨ [A, B] ψ, ψ ⟩$ and the product $Δ_{ψ} A Δ_{ψ} B$ are read in the product unit $u_{A} u_{B}$ (Preparation variance and spread of an assigned sharp observable clause (d)).

Proof

technique · instantiate the mathematical Schroedinger-Robertson bound at the two self-adjoint operators of the assigned observables and read the definitions of mean and spread from the moment theorem

Given: The data of the statement and [F1]-[F4].

1.1givenF1F3

[Weak form] Let $ψ \in D (A) \cap D (B)$ be a unit vector. By [F1] the means $a = ⟨ A ψ, ψ ⟩$ and $b = ⟨ B ψ, ψ ⟩$ are real, the spreads $Δ_{ψ} A$ and $Δ_{ψ} B$ are defined, and they equal the norms of the centered vectors $x = (A - a) ψ$ and $y = (B - b) ψ$ . By [F3], $Im ⟨ x, y ⟩ = Im ⟨ A ψ, B ψ ⟩$ and $Δ_{ψ} A Δ_{ψ} B \geq ∣ Im ⟨ A ψ, B ψ ⟩ ∣$ , which is (a).

2.1step 1.1F1F2

[Robertson form] Assume in addition $ψ \in D (A B) \cap D (B A)$ . Then $A, B$ satisfy the product-domain hypothesis of [F2] at $u = ψ$ ; combining the Robertson bound and the anticommutator refinement of [F2] with the identifications of [F1] gives the two displays of (b), and the equality clause transfers because the centered vectors there are exactly $(A - a) ψ$ and $(B - b) ψ$ . The pairing $⟨ [A, B] ψ, ψ ⟩$ is purely imaginary by [F2]. This is (b).

3.1step 1.1step 2.1F1F2

[Bounded case] If $A$ and $B$ are bounded everywhere-defined self-adjoint operators, then $D (A) = D (B) = H$ and all products are everywhere defined, so the hypotheses of clauses (a) and (b) hold for every unit $ψ \in H$ ; clauses (a) and (b) apply verbatim. This is (c).

4.1step 1.1step 2.1step 3.1F1F4∎

[Units and scope] The units are those of [F4]; the preparation spread is the outcome-law spread of Preparation variance and spread of an assigned sharp observable and no apparatus, disturbance, joint-law or time quantity enters steps 1.1-2.1, which is the scope statement (d).

The Robertson relation for preparation spreads of assigned sharp observables

Statement

Facts & Assumptions

Proof

Depends on

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