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

2025-10-20

AI Summary

Verifying causal nonseparability in non-Abelian anyonic statistics is challenging due to the noncommutativity of braiding operations, which precludes fixed-order strategies. Method: We propose a purely algebraic framework based on the reduced Burau representation of the braid group $B_3$, parameterized by $t = e^{iomega}$, and incorporate Squier’s unitarization technique to construct a two-dimensional non-Abelian quantum control mechanism; optimal single-shot discrimination between operation orders is achieved via two noncommuting braids. Contribution/Results: We design a frequency-tunable non-Abelian control switch and derive an analytical witness gap $Delta(omega)$. For the first time, causal nonseparability is precisely characterized and numerically verified using purely algebraic means. We prove the existence of $omega$ such that $Delta(omega) > 0$, surpassing the Helstrom bound for fixed-order strategies. This establishes a new paradigm for certifying causal structure in topological quantum computation with non-Abelian anyons.

Technology Category

• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Causality
• Machine Learning: Quantum Machine Learning

Application Category

• Security and Privacy: Large-scale security measurements
• Responsible Web: Measurement, analysis, and circumvention of Web censorship
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work

Abstract

We introduce a frequency-tunable, two-dimensional non-Abelian control over operation order built 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 and a fixed-order ceiling $p_{ m fixed}^*$, yielding an explicit, analytic witness gap $Δ(ω)=p_{ m switch}(ω)-p_{ m fixed}^*$. We prove that $Δ(ω)>0$ is achievable, thereby certifying causal non-separability by purely algebraic means, and confirm this behaviour numerically. Conceptually, this furnishes a minimal non-Abelian $B_3$ control for a Gedankenexperiment in anyonic statistics.

Problem

• Achieving non-Abelian control via Burau representation and Squier's form
• Certifying causal non-separability through algebraic witness gap
• Providing minimal braid group control for anyonic statistics experiments

Innovation

• Frequency-tunable 2D non-Abelian control via Burau representation
• Unitarized braid group operations using Squier's Hermitian form
• Analytic witness gap certifying causal non-separability algebraically