LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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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.

A complete positive family consists of effects

Statement

For a nonzero finite-dimensional complex-inner-product space $H$ , if $E_{1}, \dots, E_{n}$ is a finite complete positive family as defined in Finite positive effect family, then each $E_{a}$ is an effect: $E_{a}$ and $I_{H} - E_{a}$ are non-negative.

Facts & Assumptions

Given: A nonempty finite family $E_{1}, \dots, E_{n}$ of non-negative endomorphisms on $H$ with $\sum_{a = 1}^{n} E_{a} = I_{H}$ .

[F1]

Finite positive effect family: a finite complete positive family has $E_{b}$ non-negative for every $b$ and its finite sum is $I_{H}$ ; an effect is a non-negative $E$ whose complement $I_{H} - E$ is non-negative.

[F2]

Non-negative and positive operators: an operator is non-negative when it is self-adjoint and its quadratic form is nonnegative on every vector.

Proof

1.1givenF1F2algebra

Fix $a \in {1, \dots, n}$ . The family identity gives $I_{H} - E_{a} = \sum_{b \neq a} E_{b}$ . For every $x \in H$ , each $⟨ E_{b} x, x ⟩$ is a nonnegative real by [F2]; their finite sum is therefore nonnegative, including the empty sum $0$ when $n = 1$ . The sum is self-adjoint because each $E_{b}$ is self-adjoint and the sum is finite. Thus $I_{H} - E_{a}$ is non-negative.

2.1F1step 1.1∎

Since $E_{a}$ is non-negative by [F1] and its complement is non-negative by step 1.1, it is an effect by the definition in [F1]. The index $a$ was arbitrary.

Depends on

Used by

Finite POVMs and their outcome law Definition
Every finite state ensemble is obtainable from a purification Theorem

Dependency tree · two levels

5 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, Chapters 1–3 (standard reference, not scraped)
Ronald de Wolf, Quantum Computing: Lecture Notes, §15.1 (standard reference, not scraped)