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

The gist The resulting switch admits a closed expression for the single-shot Helstrom success probability and a fixed-order ceiling pfixed, defining the fixed-order ceiling pfixed and the witness gaps ∆sw(ω) = pswitch(ω) − pfixed and ∆test(ω) = ptest(ω) − pfixed.

Braid Group Control

The study introduces a representation-theoretic model of coherent order control based on the reduced Burau representation ψ(w) of B3, specialized at t = e iω and rendered unitary by Squier’s Hermitian form [11, 2510.18186]. This framework constructs a control mixer M(ω) which is a 2×2 unitary obtained from a braid word w, and the overall switch combines two non-commuting target operations A, B ∈ U(2) through a controlled superposition of orders [1]. The minimal non–Abelian braid group B3 has two independent generators σ1, σ2 obeying the Yang–Baxter relation σ1σ2σ1 = σ2σ1σ2, and admits non–trivial two-dimensional unitary representations that are matrix– rather than phase–valued, marking the transition from Abelian order interference to genuine non-Abelian braiding [2].

Squier’s Unitary Form and Positivity Window

Squier’s modification of the reduced Burau representation is defined by β: B3 −→ GL2, Z[s, s−1] (4). Squier exhibits the Hermitian form J(s) = (s + s−1) I2 − 0 1 1 0! = s + s−1 −1 −1 s + s−1! and proves the identities βi(s)∗J(s) βi(s) = J(s) (6). After specialization at s = e iω/2, equation (6) gives us ordinary J(ω)–unitarity [5]. The form J(ω) is positive definite on the open set omega+ = (0, 2π/3) ∪ (4π/3, 2π), with boundary points ω ∈ Z corresponding to det J(ω) = 0 [11]. On any connected component of this positivity region where J(ω) is positive definite, the Euclidean unitary specialization U(ω):= R(ω) β(w) s=e iω/2 R(ω −1 ∈ U2, which is unitary because β(w)†J(ω)β = J[12].

Switch and Test Device Construction

Given a braid word w ∈ B3, the switch is defined as S(θ) = 0⟩⟨0 ⊗ BA + e iθ 1⟩⟨1 ⊗ AB, where θ = θ(ω) is an arbitrary phase map [14]. The full test device is T(ω) = Mpost(ω) ⊗ I S(θ(ω)) Mpre(ω) ⊗ I, where we allow distinct braid words to produce Mpre(ω) and Mpost(ω)[15]. The Helstrom formula for unitaries states that the optimal singleshot success is p∗ (U0, U1) = 1/2 + sinδ 2/2, where δ ∈ [0, π] is the shortest eigenphase arc (spectral spread) of U†0U1 [9]. The fixed-order ceiling is defined as p∗fixed:= max[p∗(I, AB), p∗(I, BA)] (17).

Non-Abelian Interference Contrast

The witness gaps are defined as ∆sw(ω):= pswitch(ω) − pfixed and ∆test(ω):= ptest(ω) − pfixed [19]. The interference contrast is defined as ∆int(ω):= ∆test(ω) − ∆sw(ω), where this contrast takes both positive (constructive) and negative (destructive) values across the Squier positivity region, a hallmark of matrix-valued (nonAbelian) order control [13]. This distinction between constructive and destructive regimes draws a sharp distinction between Abelian (phase-only) and non-Abelian (matrix-valued) order control [13].

Main Results

Theorem 7.1 states that for the switch device S(θ(ω)) with θ(ω) = ω and target unitaries A, B defined above, the witness gap ∆sw(ω) is strictly positive for all ω ∈ omega+ [12]. For the full test device T(ω), the interference contrast ∆int(ω) = ∆test(ω) − ∆sw(ω) takes both positive and negative values as a function of ω, signaling destructive non-Abelian interference even while ∆test(ω) may remain nonnegative [12]. This demonstrates that the control mixer M(ω) ̸= I embodies the minimal nonAbelian structure of B3: two non-commuting generators acting, after unitarization on omega+, on a two-dimensional control space [13].

Single-qubit Rotations and Target Operations

The target unitaries are set as A = Rx(1.5), B = Rz(0.75) so that [A, B] ≠ 0 while both remain simple SU(2) rotations [25]. The switch phase is fixed as θ(ω) = ω [26]. Numerical simulations confirm both enhancement and suppression regimes, establishing a minimal B3 braid control that reproduces the characteristic interference pattern expected from a Gedankenexperiment in anyonic statistics [13]. The non-commutativity of the braid generators is reflected in the noncommutativity of their unitarized specializations on omega+, which is the defining feature of non-Abelian anyons [13].

Summary

The paper introduces a representation-theoretic model based on the reduced Burau representation of B3, specialized at t = e iω and rendered unitary by Squier’s Hermitian form, to create a frequency–tunable, two–dimensional non-Abelian control of operation order [1]. This construction allows for the quantification of interference contrast ∆int(ω) = ∆test(ω) − ∆sw(ω), which reveals how non-Abelian mixers can either enhance or suppress the bare switch advantage [13]. The key finding is that across the Squier positivity region, this contrast takes both positive (constructive) and negative (destructive) values, distinguishing matrix-valued order control from Abelian phase control [13]. This provides a concrete mathematical route to detect non-Abelian exchange statistics through order-sensitive interference by showing that the non-commutativity of braid generators results in an effective superposition producing a matrix holonomy rather than a scalar phase [13]. The construction establishes that the existence of both constructive and destructive regimes for ∆int(ω) demonstrates enhancement vs. suppression relative to the bare switch, confirming causal non-separability through ∆sw(ω) > 0 while highlighting the non-Abelian nature via sign changes in ∆int(ω) [12]. The entire construction yields a minimal B3 braid control that reproduces the characteristic interference pattern expected from a Gedankenexperiment in anyonic statistics [13].

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Burau representation, Squier’s form, and non-Abelian anyons Alexander Kolpakov February 23, 2026 Abstract We introduce a frequency–tunable, two–dimensional non-Abelian control of operation order constructed from the reduced Burau representation of the braid group B3, 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 pfixed, defining the fixed-order ceiling pfixed and the witness gaps ∆sw(ω) = pswitch(ω) − pfixed and ∆test(ω) = ptest(ω) − pfixed. The non-Abelian mixers can either enhance or suppress the bare switch advantage, which we quantify by the interference contrast ∆int(ω):= ∆test(ω) − ∆sw(ω). Across the Squier positivity region, ∆int(ω) takes both positive (constructive) and negative (destructive) values, a hallmark of matrix-valued (nonAbelian) order control, while ∆sw(ω) > 0 certifies algebraic causal non-separability. Numerical simulations confirm both enhancement and suppression regimes, establishing a minimal B3 braid control that reproduces the characteristic interference pattern expected from a Gedankenexperiment in anyonic statistics. Keywords: indefinite causal order; quantum switch; braid group B3; reduced Burau representation; Squier Hermitian form; unitarization; non-Abelian anyons; Helstrom discrimination; causal witnesses PACS: 03.67.-a 03.65.Ta 03.65.Vf 02.20.Uw 05.30.Pr arXiv:2510.18186v4 [quant-ph] 20 Feb 26

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1 Introduction Indefinite causal order (ICO) is typically modeled by the quantum switch- a coherent superposition of two operation orders, AB and BA [1, 2]. Most certifications of ICO employ the process-matrix framework and causal witnesses formulated as semidefinite programs (SDPs) [3, 4].

1. Bold control over causal order discrimination: The improved system can perform single-shot binary discrimination between two quantum operations (AB and BA) with a success probability of p∗(U0, U1) = 1/2 + sin δ 2 where δ is the shortest eigenphase arc, leveraging the structure derived from the Squier representation.

2. Quantification of non-Abelian control effects: The system can distinguish between constructive and destructive non-Abelian modulation by calculating the interference contrast ∆int(ω):= ∆test(ω) − ∆sw(ω), allowing it to quantify whether non-Abelian mixers can either enhance or suppress the bare switch advantage.

3. Real-time analysis of control parameter dependence: The system can analyze how non-Abelian mixers affect distinguishability by observing that the interference contrast ∆int(ω) = ∆test(ω) − ∆sw(ω) takes both positive (constructive) and negative (destructive) values as the frequency parameter ω varies across the Squier positivity region.

4. Certification of causal non-separability: The improved system can establish algebraic causal non-separability by observing that "∆sw(ω) > 0 certifies algebraic causal non-separability, which is a condition that ensures the switch process cannot be expressed as a convex mixture of fixed-order operations."