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The Born outcome law: probability measure, mixtures, atoms and spectral support

Physical scope

Conditional consequences of the adopted observable and probability postulate P2 for the sharp observables of the framework: the outcome law of an assigned self-adjoint observable and preparation is a probability measure on the Borel sets; it is affine under statistical mixtures; it assigns zero to an exact value that is not an eigenvalue; it is supported in the spectrum of the observable; and it carries no claimed density. The item is conditional on P1 and P2 and on the choice hypotheses of the spectral theorem; it asserts nothing about concrete apparatuses, joint measurements, moments, state updates or empirical data.

Statement

Adopt the state and composition postulate State and composition postulate P1: Hilbert spaces, preparations and composite systems (P1) and the observable and probability postulate Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes (P2), and assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice), the choice hypotheses of the spectral suppliers. Let $H$ be the state space assigned to a declared system, let $(A, u_{A})$ be an assigned sharp observable of the framework with spectral projection-valued measure $E_{A}$ and spectrum $σ (A)$ , and let $μ_{ρ}^{A}$ be the outcome law of Assigned sharp observables and the outcome law of the framework. Then for every preparation $ρ$ (The state space of the framework: preparations, pure rays and density operators) and every Borel set $B \subseteq R$ :

(a) Probability measure. $0 \leq μ_{ρ}^{A} (B) \leq 1$ and $μ_{ρ}^{A} (R) = 1$ ; for pairwise disjoint Borel sets $B_{n}$ with union $B$ one has $μ_{ρ}^{A} (B) = \sum_{n} μ_{ρ}^{A} (B_{n})$ ; and for the pure preparation of a unit vector $ψ$ the law is $μ_{ψ}^{A} (B) = ∥ E_{A} (B) ψ ∥^{2}$ . The declared outcome law of the model is therefore a probability measure on the Borel $σ$ -algebra of $R$ , not a formal weight attached to eigenvectors.

(b) Mixtures. If $ρ = \sum_{n} w_{n} ρ_{n}$ is a statistical mixture of preparations with weights $w_{n} \geq 0$ , $\sum_{n} w_{n} = 1$ , then $μ_{ρ}^{A} (B) = \sum_{n} w_{n} μ_{ρ_{n}}^{A} (B)$ for every Borel $B$ , the series converging absolutely: the outcome law of a mixture is the weighted average of the outcome laws, with classical weights.

(c) Exact values. For $λ \in R$ the probability of the exact value $λ$ is $μ_{ρ}^{A} ({λ}) = tr (ρ E_{A} ({λ}))$ ; it is zero for every preparation whenever $λ$ is not an eigenvalue of $A$ , and when $λ$ is an eigenvalue it equals the trace weight of the eigenspace $\ker (A - λ)$ , with $μ_{T_{u, u}}^{A} ({λ}) = 1$ for every unit eigenvector $u$ .

(d) Spectral support. $μ_{ρ}^{A} (B) = μ_{ρ}^{A} (B \cap σ (A))$ and $μ_{ρ}^{A} (σ (A)) = 1$ : outcomes outside the spectrum of the observable have probability zero.

(e) No density and no discreteness are asserted. The postulate asserts that $μ_{ρ}^{A}$ is a probability measure on the Borel sets and does not assert that it is a density with respect to Lebesgue measure or a finite or countable sum of point masses; the operational object named there is the probability of a declared Borel set, such as a finite-resolution bin, not the value of a density at a point (Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes clause (iii)). Which structure a particular pair $(A, ρ)$ exhibits is a property of that pair and is not fixed by the adopted law.

(f) What is not claimed. No mean, variance or higher moment is asserted by this item; their existence is conditional and is treated separately. No claim is made about the joint statistics of two or more observables, about the state after the outcome, about the physical mechanism producing an outcome, or about any concrete apparatus. The statement is conditional on P1 and P2 and on the cited mathematics.

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); the assigned state space $H$ with preparation class $D (H)$ of The state space of the framework: preparations, pure rays and density operators; an assigned sharp observable $(A, u_{A})$ with spectral PVM $E_{A}$ and spectrum $σ (A)$ ; preparations $ρ, ρ_{n}$ ; Borel sets $B, B_{n} \subseteq R$ ; and the readings of Assigned sharp observables and the outcome law of the framework.

[F1]

P1. Pure preparations are the unit rays of the assigned Hilbert space, general preparations are the density operators $ρ$ with $tr ρ = 1$ , and a statistical mixture $\sum_{n} w_{n} ρ_{n}$ with $w_{n} \geq 0$ , $\sum_{n} w_{n} = 1$ is a preparation (State and composition postulate P1: Hilbert spaces, preparations and composite systems clauses (i)-(iii), The state space: density operators, pure and mixed states, and purity).

[F2]

P2. An ideal sharp real-valued quantity is a self-adjoint operator with a declared unit; its spectral PVM assigns to a Borel set $B$ the probability $μ_{ρ}^{A} (B) = tr (ρ E_{A} (B))$ , equal to $∥ E_{A} (B) ψ ∥^{2}$ for a unit pure preparation; the law is a measure and not necessarily a density or a discrete sum; an exact continuum value is an idealization, while a real detector's finite-resolution outcome is represented by a declared Borel set (Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes clauses (i)-(iii)).

[F3]

Scalar law. For every density operator $ρ$ with expansion $ρ = \sum_{n} w_{n} T_{u_{n}, u_{n}}$ and every PVM $E$ , the assignment $B \mapsto tr (ρ E (B))$ is a probability measure with $tr (ρ E (B)) = \sum_{n} w_{n} ∥ E (B) u_{n} ∥^{2}$ ; for a unit vector $u$ it equals $∥ E (B) u ∥^{2}$ (Density operators and scalar spectral probability clauses (a) and (b)).

[F4]

Atoms and eigenvalues. $E_{A} ({λ})$ is the projection onto $\ker (A - λ)$ ; it is nonzero exactly when $λ$ is an eigenvalue of $A$ ; consequently $tr (ρ E_{A} ({λ})) = 0$ for every preparation when $λ$ is not an eigenvalue, and $μ_{T_{u, u}}^{A} ({λ}) = 1$ for every unit eigenvector $u$ at $λ$ (Atoms of the spectral measure and eigenvalues of a self-adjoint operator clauses (a) and (b)).

[F5]

Spectral support. $E_{A} (R ∖ σ (A)) = 0$ and $tr (ρ E_{A} (B)) = tr (ρ E_{A} (B \cap σ (A)))$ with $μ_{ρ}^{A} (σ (A)) = 1$ for every preparation (Support of the spectral measure, the spectrum and the values of the scalar law clauses (a) and (b)).

[F6]

Affinity. For a trace-norm convergent mixture $ρ = \sum_{n} w_{n} ρ_{n}$ of density operators one has $tr (ρ E_{A} (B)) = \sum_{n} w_{n} tr (ρ_{n} E_{A} (B))$ for every Borel set $B$ , the series converging absolutely (Affinity and trace-norm continuity of the scalar spectral law clauses (b) and (c)).

[F7]

Nature of the law (P2 clause (iii)). The postulate asserts that the outcome law is a measure, not necessarily a density with respect to Lebesgue measure and not necessarily a finite or countable sum of point masses; an exact value of a continuum quantity is an idealization, and a real detector reports the probability of a declared Borel set such as a bin (Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes clause (iii)).

[F8]

Spectral theorem. The spectral PVM $E_{A}$ exists, is unique among regular PVMs on the Borel sets of $R$ representing $A$ , and is the object whose scalar measures enter [F3] (Spectral theorem for unbounded self-adjoint operators (PVM form)).

Proof

1.1givenF2F3F8algebra

[Clause (a)] By [F2] the outcome law is defined using the spectral PVM of [F8], and by $μ_{ρ}^{A} (B) = tr (ρ E_{A} (B))$ ; by [F3] it is a probability measure on the Borel sets with values in $[0, 1]$ , normalization $μ_{ρ}^{A} (R) = tr ρ = 1$ and countable additivity, and for a unit vector $ψ$ it equals $∥ E_{A} (B) ψ ∥^{2}$ . This is clause (a).

1.2givenF1F6algebra

[Clause (b)] Let $ρ = \sum_{n} w_{n} ρ_{n}$ be a statistical mixture; by [F1] it is a preparation, and [F6] gives $μ_{ρ}^{A} (B) = tr (ρ E_{A} (B)) = \sum_{n} w_{n} tr (ρ_{n} E_{A} (B)) = \sum_{n} w_{n} μ_{ρ_{n}}^{A} (B)$ for every Borel $B$ with absolute convergence.

1.3givenF4algebra

[Clause (c)] By [F4], $E_{A} ({λ})$ is the projection onto $\ker (A - λ)$ and vanishes exactly when $λ$ is not an eigenvalue; hence $μ_{ρ}^{A} ({λ}) = tr (ρ E_{A} ({λ})) = 0$ for every preparation in that case, while for a unit eigenvector $u$ at $λ$ the same reference gives $μ_{T_{u, u}}^{A} ({λ}) = 1$ , and the value is the trace weight of the eigenspace.

1.4givenF5algebra

[Clause (d)] By [F5], $E_{A} (R ∖ σ (A)) = 0$ and hence $μ_{ρ}^{A} (B) = tr (ρ E_{A} (B)) = tr (ρ E_{A} (B \cap σ (A))) = μ_{ρ}^{A} (B \cap σ (A))$ , with $μ_{ρ}^{A} (σ (A)) = tr ρ = 1$ ; this is the spectral-support reading of [F2].

1.5givenF2F7

[Clause (e)] By [F2] the law is asserted to be a probability measure, and by [F7] the postulate neither asserts that it is a density with respect to Lebesgue measure nor that it is a finite or countable sum of point masses, the operational object being the probability of a declared Borel set. Therefore clause (e) is a reading of P2 clause (iii) alone, and no further structure of $μ_{ρ}^{A}$ is claimed.

2.1step 1.1step 1.2step 1.3step 1.4step 1.5∎

[Clause (f) and conclusion] Steps 1.1-1.4 prove clauses (a)-(d) from P1, P2 and the cited mathematics, and step 1.5 records the limits of clause (e). Nothing in the argument uses moments, joint statistics, state updates, apparatus models or data, so none of these is claimed.

The Born outcome law: probability measure, mixtures, atoms and spectral support

Statement

Facts & Assumptions

Proof

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