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.

QC24 Projective Chord Distance

Definition

For an integer $D \geq 1$ and unit vectors $u, v \in C^{D}$ , write $[u]$ and $[v]$ for their pure-state rays and define the projective chord distance by

$d_{c h} ([u], [v]) = \inf_{∣ λ ∣ = 1} ∥ u - λ v ∥_{2} .$

Here $∥ \cdot ∥_{2}$ is the norm induced by the root inner product, which is linear in its first argument. The infimum is over unit complex scalars, so it removes only the common phase of a state vector. The companion lemma Projective Chord Distance Formula ↗ evaluates this infimum, proves it is attained, gives the closed formula, and proves independence from the chosen unit representatives.

Depends on

Used by

QC24 State Preparation Cost Model Definition
Projective Chord Distance Formula Lemma
QC24 Signed Ray Packing Code Lemma
Amplitude Table Precision and Compilation Theorem
QC24 Discrete Alphabet Approximate Preparation Lower Bound Theorem

Dependency tree · two levels

7 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 Preskill, Quantum Computation, Chapter 5, Classical and Quantum Circuits (standard reference, not scraped)
John Watrous, The Theory of Quantum Information (standard reference, not scraped)