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Burau representation, Squier's form, and non-Abelian anyons

Alexander Kolpakov

TL;DR

This work develops a minimal, fully unitary realization of non-Abelian indefinite causal order using the reduced Burau representation of the braid group $B_3$, specialized at $t=e^{i\omega}$ and unitarized by Squier's Hermitian form. A two-dimensional mixer derived from a $B_3$ word coherently couples two non-commuting qubit unitaries, enabling a switch whose Helstrom-discrimination probability yields an analytic witness gap $\Delta(\omega)$ that certifies causal non-separability when positive. The authors derive exact conditions under which the construction remains unitary within the Squier positivity window, and they demonstrate, via a concrete example with single-qubit rotations, that the switch exhibits a positive enhancement gap while a non-Abelian test device can show a sign change, reflecting Abelian versus non-Abelian order interference. This provides a mathematical, substrate-independent Gedankenexperiment for detecting non-Abelian exchange statistics through order-sensitive interference, with reproducible code and figures available publicly.

Abstract

We introduce a frequency-tunable, two-dimensional non-Abelian control of operation order constructed from the reduced Burau representation of the braid group $B_3$, specialised at $t=e^{iω}$ and unitarized by Squier's Hermitian form. Coupled to two non-commuting qubit unitaries $A,B$, the resulting switch admits a closed expression for the single-shot Helstrom success probability and a fixed-order ceiling $p_{\mathrm{fixed}}$, defining the analytic witness gap $Δ(ω)=p_{\mathrm{switch}}(ω)-p_{\mathrm{fixed}}$. The sign change of $Δ(ω)$ across the Squier positivity window reveals alternating constructive and destructive interference of causal orders, a hallmark of non-Abelian control, while $Δ(ω)>0$ certifies algebraic causal non-separability. Numerical simulations confirm both enhancement and suppression regimes, establishing a minimal $B_3$ braid control that reproduces the characteristic interference pattern expected from a Gedankenexperiment in anyonic statistics.

Burau representation, Squier's form, and non-Abelian anyons

TL;DR

This work develops a minimal, fully unitary realization of non-Abelian indefinite causal order using the reduced Burau representation of the braid group , specialized at and unitarized by Squier's Hermitian form. A two-dimensional mixer derived from a word coherently couples two non-commuting qubit unitaries, enabling a switch whose Helstrom-discrimination probability yields an analytic witness gap that certifies causal non-separability when positive. The authors derive exact conditions under which the construction remains unitary within the Squier positivity window, and they demonstrate, via a concrete example with single-qubit rotations, that the switch exhibits a positive enhancement gap while a non-Abelian test device can show a sign change, reflecting Abelian versus non-Abelian order interference. This provides a mathematical, substrate-independent Gedankenexperiment for detecting non-Abelian exchange statistics through order-sensitive interference, with reproducible code and figures available publicly.

Abstract

We introduce a frequency-tunable, two-dimensional non-Abelian control of operation order constructed from the reduced Burau representation of the braid group , specialised at and unitarized by Squier's Hermitian form. Coupled to two non-commuting qubit unitaries , the resulting switch admits a closed expression for the single-shot Helstrom success probability and a fixed-order ceiling , defining the analytic witness gap . The sign change of across the Squier positivity window reveals alternating constructive and destructive interference of causal orders, a hallmark of non-Abelian control, while certifies algebraic causal non-separability. Numerical simulations confirm both enhancement and suppression regimes, establishing a minimal braid control that reproduces the characteristic interference pattern expected from a Gedankenexperiment in anyonic statistics.
Paper Structure (15 sections, 5 theorems, 37 equations, 2 figures)

This paper contains 15 sections, 5 theorems, 37 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Manuscript contents
  3. Braid group control and the switch
  4. Braid group on three strands.
  5. Pictures.
  6. Reduced Burau representation for $B_3$.
  7. Faithfulness for $B_3$.
  8. Squier’s unitarization
  9. Relation to the standard reduced Burau.
  10. Unitary specializations and the control parameter.
  11. Switch block and test device
  12. Helstrom's formula and convexity witness
  13. Main Results
  14. Single-qubit rotations.
  15. Interpretation.

Key Result

Lemma 5.1

For any fixed braid words $w_{\mathrm{pre}/\mathrm{post}}\in B_3$, the set is a finite union of open intervals whose endpoints lie in $Z$. On each connected component $I\subset\Omega_+$ one can choose $R(\omega)$ with $J(\omega)=R(\omega)^\dagger R(\omega)$ that depends real-analytically on $\omega$, and consequently are $U(2)$–valued, real-analytic functions of $\omega$ on $I$.

Figures (2)

  • Figure 1: Two braid generators of $B_3$, $\sigma_1$ and $\sigma_2$, and their product $w = \sigma_1 \sigma_2 \sigma_1$ (which satisfies $w = \sigma_2 \sigma_1 \sigma_2$, up to strand rearrangement).
  • Figure 2: Left: Verification of Squier’s $J$-unitarity for $\beta_1, \beta_2$. Center: Euclidean metric error of $U(\omega)$ within the positivity window of $J(\omega)$. Right: Witness gap $\Delta(\omega)$, both Abelian and non-Abelian, in the region where $J(\omega)\succ0$.

Theorems & Definitions (9)

  • Lemma 5.1: Squier positivity intervals
  • proof
  • Proposition 5.2: Smoothness of the whole device
  • proof
  • Lemma 6.1: Helstrom formula for unitaries
  • proof
  • Proposition 6.2: Convexity witness
  • proof
  • Theorem 7.1