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Dense projective SU(2) image has dense determinant-one lifts

Statement

Assume ${A C}_{ω}$ . Let $q : S U (2) \to P U (2)$ be the quotient by ${\pm I}$ , and let $Γ \leq P U (2)$ be dense. Then the full inverse image $q^{- 1} (Γ)$ is dense in $S U (2)$ . Thus projective density gives determinant-one lifts arbitrarily close to any $S U (2)$ target, while the choice of a lift can differ by the central sign.

Facts & Assumptions

Given: The two-sheeted covering homomorphism $q : S U (2) \to P U (2)$ and a dense subgroup $Γ \leq P U (2)$ .

[F1]

$q : S U (2) \to P U (2)$ is a two-sheeted covering with kernel ${\pm I}$ . The projective one-qubit group is SO(3)

[F2]

By the given density hypothesis, $Γ$ meets every nonempty open subset of $P U (2)$ .

[A1]

${A C}_{ω}$ is the axiom assumed by the covering supplier. The Axiom of Countable Choice ( ${A C}_{ω}$ )

Proof

1.1givenF1choose

Let $O$ be any nonempty open subset of $S U (2)$ and choose $u \in O$ . By [F1], there is an evenly covered neighborhood $N_{0}$ of $q (u)$ and a sheet $S_{0}$ containing $u$ such that $q ∣_{S_{0}} : S_{0} \to N_{0}$ is a homeomorphism. Set $S = S_{0} \cap O$ and $N = q (S)$ . Since $S$ is open in $S_{0}$ , $N$ is open in $N_{0}$ ; the restriction $q ∣_{S} : S \to N$ remains a homeomorphism.

2.1A1F1F2step 1.1choose∎

The open set $N$ meets $Γ$ by [F2]; choose $g \in N \cap Γ$ . Its unique lift $v \in S$ lies in $O$ and satisfies $q (v) = g$ , hence $v \in q^{- 1} (Γ)$ . Thus every nonempty open $O$ meets $q^{- 1} (Γ)$ , which is exactly density in $S U (2)$ . The only stated axiom is [A1].

Depends on

Used by

The generated normalized Fibonacci lifts are dense in SU(2) Lemma

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

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)