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.

Diamond Norm

Definition

Let $H, K$ be nonzero finite-dimensional complex Hilbert spaces and let $Φ : L (H) \to L (K)$ be complex-linear. For each finite $r \geq 1$ , the induced trace norm of the amplification is $∥ Φ \otimes {id}_{C^{r}} ∥_{1 \to 1} : = \sup_{X \neq 0} \frac{∥ (Φ \otimes {id}_{C^{r}}) (X) ∥_{1}}{∥ X ∥_{1}},$ where $X \in L (H \otimes C^{r})$ and $∥ \cdot ∥_{1}$ is the finite-dimensional trace norm (Trace Norm). The diamond norm (the completely bounded trace norm) is $∥ Φ ∥_{⋄} : = \sup_{r \geq 1} ∥ Φ \otimes {id}_{C^{r}} ∥_{1 \to 1} .$ There is no factor $1 / 2$ in this map norm. The finite-dimensional ancilla theorem Diamond Norm Ancilla Dimension Bound ↗ proves the supremum is finite, is attained, and can be capped at $r = \dim H$ .

Depends on

Trace Norm

Used by

End-to-end channel and output error contract Definition
Channel Composition Hybrid Bound Theorem
Channel Diamond Norm One Theorem
Channel Tensor Error Bound Theorem
Diamond Norm Ancilla Dimension Bound Theorem

Dependency tree · two levels

4 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 (standard reference, not scraped)