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Bell Algebra without Bell Violation: CHSH Operators for Ising Anyons in the Kitaev Honeycomb Model

Author: Berkay Yüksel Sayim ORCID: 0009-0004-4993-7352 DOI (all versions): 10.5281/zenodo.21134322

Abstract

We construct four spatially separated Majorana zero modes per party on a finite Kitaev honeycomb lattice in its non-Abelian (Ising) phase and use them to realize a local, noncommuting CHSH measurement algebra. Two static fusion-parity observables per party commute exactly between parties and fail to commute within a party; on the maximally entangled fusion-parity Bell state the CHSH expectation reaches the algebraic Tsirelson value S = 2√2 = 2.828427, surviving fermion-parity superselection exactly — a statement specific to this four-MZM, measurement-only construction, not to braid-then-fuse protocols. A commuting-only control returns K = √2 ≤ 2, and the finite-size Majorana footprint leaking across the bipartition decays monotonically with system size. Following Howard and Vala [Phys. Rev. A 85, 022304 (2012)], we stress that a physical Bell-inequality violation using only topologically protected Ising-anyon operations does not follow from this construction — that step requires a non-Clifford ("magic") resource, addressed in a companion paper. A loop-threading relative phase δ reproduces the analytic corollary S(δ) = 2√2·cos²(δ/2) to a residual of 4.4×10⁻¹⁶ on the real lattice, and an independent gauge-invariant flux-holonomy diagnostic confirms no adiabatic channel mixing at the quoted scale, with the sole open discrepancy being a 0.020-rad gap between the accumulated Berry phase at one flux quantum and the target π, traced to genuine near-degeneracies along the loop rather than to a construction error.

Contents

This record is compiled from main_v1.1.tex (RevTeX 4-2).

Reproduction code and deposited data:

  • p5b_cbell_m0_reproduction.py — independent reproduction of the four locked M0 targets (P1, P2, S, K) using a mirrored Jordan–Wigner Majorana construction and a representation-independent combinatorial commutation rule (own code, not importing any part of the main construction). Prints results; values recorded in p5b_m0_targets.json.
  • p5b_m0_targets.json — the four locked targets, adversarial controls, and the lattice-realization diagnostics (Gram error, MZM footprint leak series) cited in the paper.
  • p5b_cbell_m5_lattice_sweep.py (+ dependency p5b_m3b_kitaev_realspace.py) — the actual lattice-level CHSH sweep under the loop-threading unitary, run on a finite Kitaev-honeycomb lattice (28×16, 8 vortices). This is the primary source of the residual (4.4×10⁻¹⁶) cited in the paper text.
  • p5b_cbell_m5_abstract_crosscheck.py — an independent, self-contained (lattice-free) fixed-setting reproduction of the same curve, used only as a second, unrelated cross-check (its own residual, 8.88×10⁻¹⁶, is recorded in p5b_m5_curve.json but is not the value cited in the paper text).
  • p5b_m5_curve.json — both routes above, clearly labeled, plus the loop-threading bridge diagnostics and the nonlocality threshold δ*.
  • p5b_cbell_m5_flux_holonomy.py + p5b_m5_flux.json — the gauge-invariant continuous-flux Wilson-holonomy diagnostic (Sec. "Continuous-flux holonomy: an honest partial"): near-trivial holonomy (‖[W,Z_A]‖_F = 0.0132), with the single open discrepancy (criterion c3, δ_full mod 2π = 3.1212 vs. target π) traced to genuine near-degeneracies along the flux loop.
  • p5b_m0_numeric.py + p5b_m0_finite_size.json — the M0 make-or-break test on the microscopic lattice: whether the Kitaev honeycomb grants Alice two non-commuting measurement settings locally on her own vortex region, together with the isotropic finite-size scan that re-anchors the leakage series and its monotonic-decrease check.

All scripts require NumPy (and SciPy for p5b_cbell_m5_lattice_sweep.py) and are deterministic.

License