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Finite bounded instruments normalize measurement probabilities

1 · Prerequisites

2 · Summary

This page defines finite bounded Kraus instrument data and proves their outcome effects, normalized probabilities, and conditional density-state updates for trace-class states on any nonzero complex Hilbert space, assuming Countable Choice. These are mathematical identities and do not establish physical apparatus realization or a measurement postulate. The companion page records the separate boundary between formal coefficients and analytic convergence.

Finite bounded Kraus instrument fixes the finite operator family and normalization convention. Finite bounded instruments normalize measurement probabilities proves the probability and state-update identities from the exact density-expansion and bounded trace-cyclicity suppliers.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

DefinitionDefinition: Literature-sourcedProof: Not applicableverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)Open item page →

Finite bounded Kraus instrument

Definition

Assume the Axiom of Countable Choice, written ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $H$ be a nonzero complex Hilbert space with inner product linear in its first argument (Hilbert space, Real and complex inner-product spaces and their induced length). Write $B (H)$ for the bounded linear operators on $H$ (The spaces (\mathcal B(X,Y)) and (\mathcal B(X)) of bounded linear operators); for $M \in B (H)$ , $M^{*}$ is its Hilbert adjoint, characterized by $⟨ M x, y ⟩ = ⟨ x, M^{*} y ⟩$ (The Hilbert-space adjoint of a bounded operator).

Under this same ${A C}_{ω}$ hypothesis, Hilbert-adjoint identities gives $M^{*} \in B (H)$ and $∥ M^{*} ∥ = ∥ M ∥$ .

A finite bounded Kraus instrument datum on $H$ consists of an integer $m \geq 1$ , integers $q_{a} \geq 0$ for $a = 1, \dots, m$ , and bounded operators $M_{a, r} \in B (H)$ for $r = 1, \dots, q_{a}$ . It is normalized when

$\sum_{a = 1}^{m} \sum_{r = 1}^{q_{a}} M_{a, r}^{*} M_{a, r} = I_{H} .$

Empty inner sums are zero. For each outcome $a$ , define its outcome operator and operation by

$E_{a} : = \sum_{r = 1}^{q_{a}} M_{a, r}^{*} M_{a, r}, J_{a} (X) : = \sum_{r = 1}^{q_{a}} M_{a, r} X M_{a, r}^{*} (X \in B (H)) .$

These formulas take values in $B (H)$ : the adjoint theorem above makes each $M_{a, r}^{*}$ bounded, and $∥ M_{a, r} X M_{a, r}^{*} x ∥ \leq ∥ M_{a, r} ∥ ∥ X ∥ ∥ M_{a, r}^{*} ∥ ∥ x ∥ = ∥ M_{a, r} ∥^{2} ∥ X ∥ ∥ x ∥$ ; the sums are finite. This definition specifies only the displayed finite Kraus form; it is not a representation theorem for arbitrary instruments and does not assert that an apparatus realizes these data.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)Open item page →

Finite bounded instruments normalize measurement probabilities

Statement

Assume the Axiom of Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $H$ carry a normalized finite bounded Kraus instrument datum as defined in Finite bounded Kraus instrument, and let $ρ$ be a density operator on $H$ : a self-adjoint positive trace-class operator with $tr ρ = 1$ (Density operators and scalar spectral probability, Trace class operator, Trace of a trace class operator, Self-adjoint, positive, unitary and normal operators). For each outcome $a$ , put $p_{a} : = tr (J_{a} (ρ))$ . Then each $E_{a}$ is an effect ( $E_{a} \geq 0$ and $I_{H} - E_{a} \geq 0$ ), each $J_{a} (ρ)$ is positive trace class, and

$p_{a} = tr (J_{a} (ρ)) = tr (ρ E_{a}) \in [0, 1], \sum_{a = 1}^{m} p_{a} = 1.$

The nonselective output $\sum_{a} J_{a} (ρ)$ is a density operator. If $p_{a} > 0$ , the normalized conditional state $ρ_{a} : = J_{a} (ρ) / p_{a}$ is a density operator; when $p_{a} = 0$ , this normalized update is undefined. For a unit vector $ψ \in H$ , the rank-one density operator $ρ = T_{ψ, ψ}$ has outcome probability $p_{a} = \sum_{r = 1}^{q_{a}} ∥ M_{a, r} ψ ∥^{2}$ . These are conditional mathematical identities; they do not assert that an apparatus realizes the instrument or supply a physical measurement postulate.

Facts & Assumptions

Given: Countable Choice; a nonzero complex Hilbert space $H$ ; a normalized finite bounded Kraus instrument datum $(M_{a, r})$ ; and a density operator $ρ$ on $H$ .

[A1]

Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )) is assumed for the adjoint and trace-class definitions and suppliers, and for the density expansion [F1]. No stronger choice principle is invoked.

[L1]

Finite bounded Kraus instrument: the outcome operators are $E_{a} = \sum_{r} M_{a, r}^{*} M_{a, r}$ , the maps are $J_{a} (X) = \sum_{r} M_{a, r} X M_{a, r}^{*}$ , and normalization means $\sum_{a, r} M_{a, r}^{*} M_{a, r} = I_{H}$ . The adjoint obeys $⟨ M x, y ⟩ = ⟨ x, M^{*} y ⟩$ .

[F1]

Density operators and scalar spectral probability: under Countable Choice, a density operator is self-adjoint, positive and trace class with trace one, and has a trace-norm convergent expansion $ρ = \sum_{j} w_{j} T_{u_{j}, u_{j}}$ with $(u_{j})$ orthonormal, $w_{j} \geq 0$ and $\sum_{j} w_{j} = 1$ . We use statement (a); its separate statement (c) is not used.

[F2]

Nuclear bounded cyclicity for trace-class operators: under Countable Choice, bounded multiplication of a nuclear representation gives trace-norm-convergent series for $B T$ and $T B$ , with $tr (B T) = \sum_{j} ⟨ B u_{j}, v_{j} ⟩$ , $tr (T B) = \sum_{j} ⟨ u_{j}, B^{*} v_{j} ⟩$ , absolute convergence, and $tr (B T) = tr (T B)$ . No unbounded factor is cyclically permuted.

[F3]

Rank-one operator calculus: $T_{u, v} x = ⟨ x, v ⟩ u$ is trace class with $∥ T_{u, v} ∥_{1} = ∥ u ∥ ∥ v ∥$ and $tr (T_{u, v}) = ⟨ u, v ⟩$ ; $T_{x, x}$ is self-adjoint and positive.

[F4]

Rank-one operator calculus: for bounded $B$ , $B T_{u, v} = T_{B u, v}$ and $T_{u, v} B = T_{u, B^{*} v}$ .

[F5]

Trace class operator: a trace-class operator is in $B (H)$ and has finite trace norm. Trace of a trace class operator defines the trace relative to a supplied Hilbert basis. Trace is absolutely convergent and basis independent proves basis independence, linearity and $∣ tr T ∣ \leq ∥ T ∥_{1}$ , hence continuity in trace norm.

[F6]

Self-adjoint, positive, unitary and normal operators: positivity means that the quadratic-form values $⟨ T x, x ⟩$ are real and nonnegative; self-adjointness is the separate identity $T^{*} = T$ .

[F7]

Trace class is a two sided Banach operator ideal: under Countable Choice, if $T$ is trace class and $A, B$ are bounded, then $A T B$ is trace class and $∥ A T B ∥_{1} \leq ∥ A ∥ ∥ T ∥_{1} ∥ B ∥$ .

[F8]

Hilbert-adjoint identities: under ${A C}_{ω}$ , the Hilbert adjoint of every bounded operator $M$ is bounded, $∥ M^{*} ∥ = ∥ M ∥$ , and $M^{* *} = M$ .

Proof

1.1givenL1F6F8algebra

For each $a, r$ , $M_{a, r}^{*} M_{a, r}$ is bounded by [F8]. Also [F8] gives $M_{a, r}^{* *} = M_{a, r}$ , so $⟨ M_{a, r}^{*} M_{a, r} x, y ⟩ = ⟨ M_{a, r} x, M_{a, r}^{* *} y ⟩ = ⟨ M_{a, r} x, M_{a, r} y ⟩ = ⟨ x, M_{a, r}^{*} M_{a, r} y ⟩$ ; hence it is self-adjoint. Its quadratic form is $∥ M_{a, r} x ∥^{2} \geq 0$ . Hence $E_{a} \geq 0$ . Normalization gives $I_{H} - E_{a} = \sum_{b \neq a} \sum_{r} M_{b, r}^{*} M_{b, r} \geq 0$ , including empty sums, so $E_{a}$ is an effect.

1.2givenA1L1F1F6F7F8algebra

By [F8], each $M_{a, r}^{*}$ is bounded, so [F7] makes $M_{a, r} ρ M_{a, r}^{*}$ trace class; the finite sum $J_{a} (ρ)$ is trace class. By [F8], $M_{a, r}^{* *} = M_{a, r}$ , and for all $x, y \in H$ , $⟨ M_{a, r} ρ M_{a, r}^{*} x, y ⟩ = ⟨ ρ M_{a, r}^{*} x, M_{a, r}^{*} y ⟩ = ⟨ M_{a, r}^{*} x, ρ M_{a, r}^{*} y ⟩ = ⟨ x, M_{a, r}^{* *} ρ M_{a, r}^{*} y ⟩ = ⟨ x, M_{a, r} ρ M_{a, r}^{*} y ⟩$ , where the first equality uses the adjoint identity for $M_{a, r}$ , the second uses $ρ = ρ^{*}$ , and the fourth uses the adjoint identity for $M_{a, r}^{*}$ . Also $⟨ M_{a, r} ρ M_{a, r}^{*} x, x ⟩ = ⟨ ρ M_{a, r}^{*} x, M_{a, r}^{*} x ⟩ \geq 0$ . Finite summation shows $J_{a} (ρ)$ is self-adjoint and positive.

2.1givenA1L1F1F2F3F4F5F7F8step 1.2algebra

By [F1], write $ρ = \sum_{j} w_{j} T_{u_{j}, u_{j}}$ in trace norm. For its partial sums $ρ_{N} = \sum_{j \leq N} w_{j} T_{u_{j}, u_{j}}$ , [F8] gives $∥ M_{a, r}^{*} ∥ = ∥ M_{a, r} ∥ < \infty$ , so [F7] gives $∥ M_{a, r} (ρ - ρ_{N}) M_{a, r}^{*} ∥_{1} \leq ∥ M_{a, r} ∥ ∥ ρ - ρ_{N} ∥_{1} ∥ M_{a, r}^{*} ∥ \to 0$ . Thus [F4] gives $J_{a} (ρ) = \sum_{r, j} w_{j} T_{M_{a, r} u_{j}, M_{a, r} u_{j}}$ in trace norm; the trace-norm sum is finite since $\sum_{j} w_{j} ∥ M_{a, r} u_{j} ∥^{2} \leq ∥ M_{a, r} ∥^{2} \sum_{j} w_{j} = ∥ M_{a, r} ∥^{2}$ for each $r$ . Trace continuity [F5] and [F3] yield $p_{a} = \sum_{r, j} w_{j} ∥ M_{a, r} u_{j} ∥^{2} \geq 0$ . Cyclicity [F2] gives $p_{a} = \sum_{r} tr (ρ M_{a, r}^{*} M_{a, r}) = tr (ρ E_{a})$ .

3.1givenA1L1F1step 2.1algebra

Because the outcome and Kraus index sets are finite, [F1] and step 2.1 give $\sum_{a} p_{a} = \sum_{j} w_{j} ⟨ (\sum_{a, r} M_{a, r}^{*} M_{a, r}) u_{j}, u_{j} ⟩ = \sum_{j} w_{j} ∥ u_{j} ∥^{2} = 1$ by normalization. Since every $p_{a} \geq 0$ , this also gives $p_{a} \leq 1$ for every $a$ .

4.1givenA1L1step 1.2step 2.1step 3.1F1F5algebra

If $p_{a} > 0$ , step 1.2 gives a positive trace-class numerator $J_{a} (ρ)$ and step 2.1 gives its trace $p_{a}$ , so $ρ_{a} = J_{a} (ρ) / p_{a}$ is positive trace class with trace one, hence a density operator by [F1]. The finite sum $\sum_{a} J_{a} (ρ)$ is likewise positive trace class, and its trace is $\sum_{a} p_{a} = 1$ by step 3.1, so it is a density operator. If $p_{a} = 0$ , division by $p_{a}$ is undefined and no normalized update is asserted.

5.1givenL1F1F3F4step 2.1algebra∎

For a unit vector $ψ$ , [F3] gives that $T_{ψ, ψ}$ is self-adjoint, positive and trace class with trace $∥ ψ ∥^{2} = 1$ , hence a density operator by [F1]. Also [F4] gives $M_{a, r} T_{ψ, ψ} M_{a, r}^{*} = T_{M_{a, r} ψ, M_{a, r} ψ}$ . Taking traces and summing over the finite Kraus list yields $p_{a} = \sum_{r} ∥ M_{a, r} ψ ∥^{2}$ , as claimed.

5 · Examples, experiments and counterexamples

None yet.

Finite bounded instruments normalize measurement probabilities

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

Finite bounded Kraus instrument

Definition

Finite bounded instruments normalize measurement probabilities

Statement

Facts & Assumptions

Proof

5 · Examples, experiments and counterexamples

Sources