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The braid phase is a common scalar character

Statement

Fix the Fibonacci convention with common exchange scalar $R_{1}$ . Let $e : B_{n} \to Z$ be exponent sum, sending each Artin generator $σ_{i}$ to $1$ . If $\tilde{ρ}$ is the generator-normalized representation $\tilde{ρ} (σ_{i}) = R_{1}^{- 1} ρ (σ_{i})$ , then for every braid $b$ and every total-charge block, $ρ (b) = R_{1}^{e (b)} \tilde{ρ} (b) .$ The scalar is the same on every total-charge block and every spectator multiplicity. It is a global phase for an uncontrolled fixed-sector computation; under coherent control it is a relative phase and must be tracked.

Facts & Assumptions

Given: The fixed-chirality Fibonacci $F / R$ data and the Artin presentation.

[A1]

Assume every generator has the same scalar $R_{1}$ on all fusion channels; its relative channel phases define the normalized generator. The actual Fibonacci data carrier fixes $R_{1} = e^{- 4 π i / 5}$ and $R_{τ} = e^{3 π i / 5}$ in its stated gauge. Fibonacci fusion data and chiral convention

[F2]

Standalone unit-modulus phases leave pure-state measurement probabilities unchanged. Standalone Global Phase Leaves Pure-State POVM Probabilities Unchanged

[F3]

A separately controlled scalar phase can change measurement probabilities by becoming relative to the control branch. A Controlled Global Phase Becomes a Relative Control Phase

Proof

1.1givenalgebra

Exponent sum is well defined on the Artin group: both sides of every adjacent braid relation have sum $3$ , both sides of every distant commutation relation have sum $2$ , and inverse cancellation changes the sum by $0$ . Thus $e : B_{n} \to Z$ is a homomorphism.

2.1A1step 1.1algebra

For a word $b = σ_{i_{1}}^{ϵ_{1}} \dots σ_{i_{k}}^{ϵ_{k}}$ , each positive generator contributes $R_{1}$ and each inverse contributes $R_{1}^{- 1}$ to the scalar, by [A1]. The scalar product is therefore $R_{1}^{ϵ_{1} + \dots + ϵ_{k}} = R_{1}^{e (b)}$ , while the remaining normalized matrices multiply to $\tilde{ρ} (b)$ . Since every relation preserves exponent sum, this factorization is independent of the word representative.

3.1F2F3step 2.1∎

The same scalar multiplies every charge block, so [F2] applies to an uncontrolled isolated sector. Under coherent control the scalar multiplies only the active branch, and [F3] shows it becomes a relative phase; it cannot be discarded without recording or correcting it.

Depends on

Used by

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Sources

Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, and Sankar Das Sarma, Non-Abelian Anyons and Topological Quantum Computation (standard reference, not scraped)
John Preskill, Lecture Notes for Physics 219: Quantum Computation, Chapter 9: Topological Quantum Computation (standard reference, not scraped)