LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-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.

Projective Chord Distance Formula

Statement

For unit vectors $u, v \in C^{D}$ ,

$d_{c h} ([u], [v]) = \sqrt{2 - 2 ∣ ⟨ u, v ⟩_{r o o t} ∣} .$

In particular, this value is independent of the chosen unit representatives, and it is zero exactly when the two vectors represent the same ray.

Facts & Assumptions

Given: Unit vectors $u, v \in C^{D}$ and the projective chord distance of QC24 Projective Chord Distance.

[F1]

For unit representatives, $d_{c h} ([u], [v]) = \inf_{∣ λ ∣ = 1} ∥ u - λ v ∥_{2}$ (QC24 Projective Chord Distance).

[F2]

The root pairing is linear in the first argument, conjugate-linear in the second, conjugate-symmetric, and induces $∥ w ∥_{2}^{2} = ⟨ w, w ⟩_{r o o t}$ (Real and complex inner-product spaces and their induced length).

[F3]

Two nonzero vectors represent the same pure-state ray exactly when one is a nonzero complex scalar multiple of the other; for unit vectors that scalar has modulus one (Pure state and ray).

[F4]

Cauchy–Schwarz gives $∣ ⟨ u, v ⟩_{r o o t} ∣ \leq ∥ u ∥_{2} ∥ v ∥_{2} = 1$ for unit vectors (Cauchy–Schwarz: $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ , with equality exactly for linearly dependent vectors).

Proof

Given: As in Facts & Assumptions.

1.1F1F2algebra

Put $c = ⟨ v, u ⟩_{r o o t}$ . For every $∣ λ ∣ = 1$ , expand by [F2]: $∥ u - λ v ∥_{2}^{2} = 2 - \overline{λ} ⟨ u, v ⟩_{r o o t} - λ ⟨ v, u ⟩_{r o o t} = 2 - 2 Re (λ c)$ .

2.1F1F4step 1.1choosealgebra

If $c = 0$ , the real part is zero for every unit $λ$ ; if $c \neq 0$ , choose $λ = \overline{c} / ∣ c ∣$ , giving $Re (λ c) = ∣ c ∣$ . Since $Re (λ c) \leq ∣ c ∣$ for every unit $λ$ , this phase attains the infimum in [F1]; the value is $\sqrt{2 - 2 ∣ c ∣}$ , a real nonnegative number by [F4].

3.1F1F2F3step 2.1algebra∎

Replacing $u$ and $v$ by unit scalar multiples changes $⟨ u, v ⟩_{r o o t}$ by a unit scalar, so its modulus and the displayed value do not change. Since the infimum is attained, it is zero exactly when a minimizing phase satisfies $u = λ v$ , which is exactly equality of the represented rays by [F3].

Depends on

Used by

QC24 Signed Ray Packing Code Lemma
Amplitude Table Precision and Compilation Theorem
QC24 Discrete Alphabet Approximate Preparation Lower Bound Theorem

Cited to discharge well-definedness by QC24 Projective Chord Distance.

Dependency tree · two levels

12 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, Chapter 1 (standard reference, not scraped)
John Preskill, Quantum Computation, Chapter 5, Classical and Quantum Circuits (standard reference, not scraped)