DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generated

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.

Finite positive effect family

Definition

Let $H$ be a nonzero finite-dimensional complex inner-product space, and let $I_{H} : H \to H$ denote the identity map $x \mapsto x$ . An effect on $H$ is an endomorphism $E$ such that both $E$ and $I_{H} - E$ are non-negative in the sense of Non-negative and positive operators. A finite complete positive family is a nonempty finite list $(E_{1}, \dots, E_{n})$ of non-negative endomorphisms with $\sum_{a = 1}^{n} E_{a} = I_{H}$ . The next lemma proves that every member of such a family is an effect, so the family is a finite complete effect family. This is algebraic data only; it does not assert a measurement law.

Depends on

Used by

Dependency tree · two levels

20 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)