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.

Pure state and ray

Definition

Let $H$ be a nonzero finite-dimensional complex inner-product space. Two nonzero vectors $x, y \in H$ represent the same ray when $y = α x$ for some nonzero $α \in C$ . A pure state is such a ray. If $u$ is a unit representative, its associated operator is $P_{u} = ∣ u ⟩ ⟨ u ∣$ , with the bra, ket and outer-product convention of Dirac Notation Conversion. The theorem Ray-to-projector is well-defined and bijective ↗ proves that this operator is a rank-one orthogonal projection independent of the chosen unit representative, so it is the operator representative of the pure state.

Depends on

Used by

Coherent finite sampler and observable value Definition
Continuous-time quantum walk Definition
Eigenphase estimation access model Definition
Finite adiabatic path model Definition
Finite open-graph adaptive measurement pattern Definition
Orthogonal-Projector Reflection and Access Contract Definition
Prepared unitary and success subspace for amplitude amplification Definition
Pure-State Entanglement Across a Tensor Cut Definition
QC24 Complex Projective Space and Its Standard Patches Definition
QC24 Projective Chord Distance Definition
QC24 State Preparation Cost Model Definition
Quantum linear-systems state-output task Definition
Qubit spaces and states Definition
Relative Phase Definition
Stabilizer state Definition
State-dependent spectral distribution Definition
Bloch coordinates of |0> and H|0> Example
One-bit feed-forward and deferred measurement Example
Trace distance of orthogonal pure states Example
Adversary progress at initial and bounded-error final states Lemma
An anticommuting stabilizer measurement is fair Lemma
Deterministic Pauli measurement on a pure stabilizer state Lemma
Measurement law of a coherent sampler Lemma
One-bit J-gate teleportation branch identity Lemma
Orthogonal marked and unmarked components of the uniform state Lemma
Phase-estimation finite outcome law Lemma
Projective Chord Distance Formula Lemma
Fusion measurement branches are normalized exactly when nonzero Theorem
Pattern branch probabilities and zero-probability transcripts Theorem
Phase kickback from the standard bit query Theorem
Pure State No Cloning Theorem
Purity is the boundary of the Bloch ball Theorem
QC24 Finite Alphabet Exact Nonuniversality Theorem
Rank-one density operators are pure-state projectors Theorem
Rank-One Reflection Is Ray-Invariant and Its Signs Differ Under Control Theorem
Ray-to-projector is well-defined and bijective Theorem
Swap-test measurement and pure-state outcome law Theorem
The measurement law for a finite continuous-time walk Theorem

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

John Watrous, The Theory of Quantum Information, §2.1.2 (standard reference, not scraped)
John Preskill, Quantum Computation, Chapter 2: Foundations I: States and Ensembles (standard reference, not scraped)
Ronald de Wolf, Quantum Computing: Lecture Notes, §15.1 (standard reference, not scraped)