LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generated

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.

The projective one-qubit group is SO(3)

Statement

Assume ${A C}_{ω}$ . Give $P U (2)$ the quotient metric from Projective braid gate and operator metric. The map $q : S U (2) ⟶ P U (2), q (A) = [A],$ is a two-sheeted covering with kernel ${I, - I}$ . It induces a topological group isomorphism $S U (2) / {\pm I} ≅ P U (2) ≅ S O (3)$ .

Facts & Assumptions

Given: The projective unitary group $P U (2)$ with its quotient metric, the matrix group $S U (2)$ , and ${A C}_{ω}$ .

[F1]

The projective distance is a metric and $d_{P} ([A], [B]) = \min_{∣ z ∣ = 1} ∥ A - z B ∥_{o p}$ . Projective braid gate and operator metric

[F2]

Closed bounded subsets of finite-dimensional Euclidean space are compact. Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line

[F3]

A compact subset of a Hausdorff space is closed. In a Hausdorff space a point and a disjoint compact set, and two disjoint compact sets, have disjoint open neighbourhoods; hence every compact subset is closed, and in a compact Hausdorff space the compact subsets are exactly the closed ones

[F4]

Every unit complex number has the form $e^{i θ}$ for some real $θ$ . $t \mapsto (\cos t, \sin t)$ is a bijection from $[0, 2 π)$ onto the real unit circle

[F5]

Quaternionic conjugation gives a surjective two-sheeted covering $ρ : S U (2) \to S O (3)$ with kernel ${I, - I}$ . SU(2) to SO(3) as a covering homomorphism

[A1]

Assume ${A C}_{ω}$ , the exact choice principle used by [F5]. The Axiom of Countable Choice ( ${A C}_{ω}$ )

Proof

1.1givenF4algebra

The determinant of a unitary $A \in U (2)$ has modulus one, since $1 = \det (A^{*} A) = \overline{\det A} \det A$ . Write $\det A = e^{i θ}$ by [F4] and put $λ = e^{- i θ / 2}$ . Then $∣ λ ∣ = 1$ and $\det (λ A) = λ^{2} \det A = 1$ , so $λ A \in S U (2)$ and $q (λ A) = q (A)$ . Thus $q$ is onto.

2.1givenstep 1.1algebra

If $A \in S U (2)$ and $q (A) = [I]$ , then $A = λ I$ for some $∣ λ ∣ = 1$ . The determinant condition gives $λ^{2} = 1$ , hence $λ = \pm 1$ . Conversely $I$ and $- I$ have the same projective class. Therefore $\ker q = {\pm I}$ , and two matrices have the same $q$ -image exactly when they differ by this kernel.

2.2F1F2F3step 1.1algebra

The map $q$ is continuous: by [F1], $d_{P} (q (A), q (B)) \leq ∥ A - B ∥_{o p}$ . Identify $M_{2} (C)$ with $R^{8}$ . The equations $A^{*} A = I$ and $\det A = 1$ define a closed subset, and each matrix entry is bounded in modulus by $1$ because each column has norm $1$ . Thus $S U (2)$ is compact by [F2]. The metric space $P U (2)$ is Hausdorff by [F1]. If $F \subseteq S U (2)$ is closed, then it is compact and $q (F)$ is compact, hence closed by [F3]. It is a quotient map: if $q^{- 1} (W)$ is open, then $q^{- 1} (P U (2) ∖ W)$ is closed; surjectivity and closedness of $q$ make its image $P U (2) ∖ W$ closed, so $W$ is open.

3.1A1F3F5step 2.1step 2.2algebra∎

By [F5] and step 2.1, $q$ and $ρ$ have exactly the same fibers. There is therefore a unique bijective group homomorphism $\overset{ˉ}{ρ} : P U (2) \to S O (3)$ with $\overset{ˉ}{ρ} (q (A)) = ρ (A)$ . It is continuous because $q$ is a quotient map and $ρ = \overset{ˉ}{ρ} \circ q$ is continuous. The space $P U (2) = q (S U (2))$ is compact, while $S O (3)$ is Hausdorff as a subspace of the real $3 \times 3$ matrix space. A continuous bijection from a compact space to a Hausdorff space is a homeomorphism: images of closed sets are compact and therefore closed by [F3]. Hence $\overset{ˉ}{ρ}$ is a topological group isomorphism. Since $q$ is a quotient map and its fibers are exactly the cosets of ${\pm I}$ by step 2.1, the induced map $S U (2) / {\pm I} \to P U (2)$ is also a homeomorphism. The covering sheets for $q$ are carried from those of $ρ$ by this homeomorphism, so $q$ is a two-sheeted covering.

Depends on

Used by

Dependency tree · two levels

68 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

Michael H. Freedman, Michael J. Larsen, and Zhenghan Wang, A Modular Functor Which Is Universal for Quantum Computation (standard reference, not scraped)
John Preskill, Lecture Notes for Physics 219: Quantum Computation, Chapter 9: Topological Quantum Computation (standard reference, not scraped)