DefinitionDefinition: AI-adaptedProof: AI-generatedPipeline-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.

Relative Phase

Definition

Fix an ordered orthonormal pair $e_{0}, e_{1}$ in a finite-dimensional complex inner-product space. If a vector has nonzero coordinates $α, β$ on this pair, its relative phase on these coordinates is

$η : = \frac{β / α}{∣ β / α ∣} \in C, ∣ η ∣ = 1.$

The relative phase is undefined when either coordinate is zero. It is unchanged by multiplying the whole vector by any nonzero complex scalar. It can distinguish rays: for the unit vectors $ψ_{η} = (e_{0} + η e_{1}) / \sqrt{2}$ , distinct values of $η$ give distinct rank-one projectors and distinct rays.

Facts & Assumptions

Given: The ordered orthonormal pair and nonzero coordinates $α, β$ in the Definition.

[F1]

Real and complex inner-product spaces and their induced length: the root pairing is linear in the first argument and conjugate-linear in the second.

[L1]

Pure state and ray defines a pure state as a ray under nonzero common scalar multiplication. Its current draft definition body was read.

[L2]

Exact one-qubit gates and rotations fixes the ordered computational basis for the qubit instance. Its current draft definition body was read; its item is now present in the QC07 manifest and page; the page remains draft.

[L3]

Dirac Notation Conversion defines $∣ x ⟩ ⟨ y ∣ z = ⟨ z, y ⟩_{r o o t} x$ and the Dirac pairing. Its current draft definition body was read; the separate coordinate-matrix identity is assigned to its linked QC01 theorem.

[L4]

Dirac Notation Coordinate Identity proves that the coordinate matrix of $∣ x ⟩ ⟨ y ∣$ is $[x] [y]^{†}$ in the stated orthonormal bases. Its current draft proof was read.

Verification

1.1givenL1algebra

For any nonzero scalar $c$ , the ratio of the two coordinates of $c ψ$ is $(c β) / (c α) = β / α$ . Its unit-modulus part is therefore also unchanged, so the Definition assigns a property of the ray.

1.2givenF1L2L3L4algebra

Since $∣ η ∣ = 1$ and $e_{0}, e_{1}$ are orthonormal, $ψ_{η}$ has norm one. In the qubit instance [L2] identifies this ordered pair with $∣ 0 ⟩, ∣ 1 ⟩$ . By [L3] and [L4], its rank-one projector has matrix $\frac{1}{2} (\begin{array}{cc} 1 & \overline{η} \\ η & 1 \end{array})$ in the ordered pair.

2.1step 1.2L1algebra∎

If $η \neq η^{'}$ , the lower-left entries of these two projector matrices differ, so the projectors differ. Also, if $ψ_{η^{'}} = c ψ_{η}$ for a nonzero scalar $c$ , comparison of the $e_{0}$ coordinate forces $c = 1$ and comparison of the $e_{1}$ coordinate forces $η^{'} = η$ , a contradiction. Hence changing the relative phase can change the ray.

Depends on

Used by

Amplitude amplification with independently chosen phase rotations Algorithm
QSP signal matrix and ordered phase sequence Definition

Dependency tree · two levels

23 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

Ronald de Wolf, Quantum Computing: Lecture Notes, arXiv:1907.09415v6 (standard reference, not scraped)
John Preskill, Quantum Computation, Chapter 2: Foundations I: States and Ensembles (standard reference, not scraped)