TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Instrument outcome weights normalize

Statement

For a finite instrument $(I_{a})_{a \in Ω}$ and a density operator $ρ$ on its input space, the branch weights $p_{a} = tr (I_{a} (ρ))$ form a probability distribution on $Ω$ .

Facts & Assumptions

Given: A finite instrument $(I_{a})_{a \in Ω}$ and an input density operator $ρ$ .

[F1]

Finite instruments and conditional states: each outcome map is completely positive and $\sum_{a} tr (I_{a} (X)) = tr (X)$ for every input operator $X$ ; $p_{a} = tr (I_{a} (ρ))$ .

[F2]

Completely positive maps: complete positivity includes the $r = 1$ extension, which is the original map and therefore preserves nonnegativity.

[F5]

Density operator: on a nonzero finite-dimensional complex inner-product space, a density operator is nonnegative and has trace one.

[F3]

A non-negative operator is equivalently self-adjoint with nonnegative eigenvalues, a positive semidefinite matrix in an orthonormal basis, or an operator of the form S^*S: a nonnegative operator has an orthonormal basis in which its matrix is diagonal with nonnegative real diagonal entries.

[F4]

The basis-independent trace of an endomorphism of a finite-dimensional vector space: trace is the basis-independent matrix trace of a finite-dimensional endomorphism.

[F6]

Non-negative and positive operators: a nonnegative operator is self-adjoint and has nonnegative quadratic form on every vector.

Proof

1.1givenF1F2F3F4F5F6

For each $a$ , complete positivity [F1]–[F2] at $r = 1$ sends the nonnegative input $ρ$ from [F5] to a nonnegative $τ_{a} : = I_{a} (ρ)$ in the sense of [F6]. By [F3], $τ_{a}$ has nonnegative diagonal entries in an orthonormal basis, and [F4] therefore gives $p_{a} = tr (τ_{a}) \geq 0$ .

1.2givenF1F5

Apply the trace-preservation identity in [F1] to $X = ρ$ . It gives $\sum_{a \in Ω} p_{a} = tr (ρ) = 1$ by [F5]. Thus the nonnegative branch weights sum to one.

2.1step 1.1step 1.2algebra∎

For any $a$ , $1 - p_{a} = \sum_{b \neq a} p_{b} \geq 0$ by step 1.1, including the empty sum when $Ω = {a}$ . Hence $p_{a} \in [0, 1]$ for every outcome, and the weights form a finite probability distribution.

Depends on

Used by

One Simon Fourier-sampling sample Algorithm
Finite authenticated key-distribution protocol Definition
Finite measurement branches and classical control Definition
Heralded success and conditional verification contract Definition
Normalized output law for finite quantum query branches Definition
Oracle-only bit-query algorithm Definition
The outcome law and outputs of a measured circuit Definition
Positive-probability instrument branches are density operators Theorem
Simon query and output measurement produce a coset pair Theorem
The nonselective output of an instrument is a density operator Theorem

Cited to discharge well-definedness by Finite instruments and conditional states.

Dependency tree · two levels

24 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John Watrous, The Theory of Quantum Information (2018), §2.3.2 (standard reference, not scraped)
John Preskill, Quantum Computation, Chapter 3: Foundations II: Measurement and Evolution, §3.2.4 (standard reference, not scraped)