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.

Bloch coordinate maps

Definition

Let $Q_{2}$ be the qubit space of Qubit spaces and states, and let $ρ$ be a density operator on $Q_{2}$ in the sense of Density operator. Define its candidate Bloch coordinates in $C^{3}$ by $r_{j} (ρ) : = tr (ρ σ_{j}) (j \in {x, y, z}), r (ρ) : = (r_{x} (ρ), r_{y} (ρ), r_{z} (ρ)) .$ For any $r = (r_{x}, r_{y}, r_{z}) \in R^{3}$ , define the associated Hermitian-matrix candidate $ρ (r) : = \frac{1}{2} (I + r_{x} X + r_{y} Y + r_{z} Z) .$ Write $r \cdot σ : = r_{x} X + r_{y} Y + r_{z} Z$ and $∥ r ∥_{2} : = \sqrt{r_{x}^{2} + r_{y}^{2} + r_{z}^{2}} .$ The next theorem proves that these trace coordinates are real and that every Hermitian trace-one $2 \times 2$ matrix is uniquely $ρ (r)$ ; the positivity theorem determines which real-coordinate candidates are states.

Depends on

Used by

Exact one-qubit gates and rotations Definition
Bloch coordinates of |0> and H|0> Example
Existence and uniqueness of Bloch coordinates Theorem
Purity is the boundary of the Bloch ball Theorem
The Bloch ball is exactly the qubit density-state space Theorem

Dependency tree · two levels

18 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 2: Foundations I: States and Ensembles (updated July 2015) (standard reference, not scraped)
Ronald de Wolf, Quantum Computing: Lecture Notes, arXiv:1907.09415v6 (2026-08-27) (standard reference, not scraped)
Aram Harrow, 6.443/8.371/18.436 Quantum Information Science II, Lecture 1: Quantum states and operations (2018) (standard reference, not scraped)