DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge 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.

Broadcasting Family

Definition

Let $H$ be a nonzero finite-dimensional complex inner-product space, let $D (H)$ be its density operators (nonnegative operators of trace one; Density operator), and let $S \subseteq D (H)$ be any set, finite or infinite. A broadcaster for $S$ is one quantum channel $B : L (H) \to L (H \otimes H)$ (completely positive and trace-preserving; Finite-dimensional quantum channel) such that for every $ρ \in S$ , ${Tr}_{1} (B (ρ)) = ρ and {Tr}_{2} (B (ρ)) = ρ .$ Here ${Tr}_{1}$ and ${Tr}_{2}$ are the partial traces over the indicated output factors (Partial trace over a finite tensor factor). The joint output may be correlated; this definition does not require $B (ρ) = ρ \otimes ρ$ .

Depends on

Used by

Full Rank Pair No Broadcasting Theorem
Singular Pair No Broadcasting Theorem
State Family No Broadcasting Iff Commuting Theorem

Dependency tree · two levels

14 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

Howard Barnum, Carlton M. Caves, Christopher A. Fuchs, Richard Jozsa, and Benjamin Schumacher, Noncommuting Mixed States Cannot Be Broadcast (standard reference, not scraped)