This is a bounded four-qubit simulation study of how four pure initial-state families evolve under one frozen Kuramoto-XY Hamiltonian. It does not show that entanglement lowers a synchronisation threshold. The model is closed and unitary: it has no drive, dissipation, limit cycle, coupling scan, provider, or hardware execution.
The earlier implementation assigned each qubit a phase with
atan2(<Y_i>, <X_i>). Bell, GHZ, and W states have vanishing local transverse
Bloch vectors in this model. atan2(0, 0) nevertheless returns zero, so the old
code assigned every site the same artificial phase and reported (R=1).
The corrected local complex amplitude is
[ z_i = \langle X_i \rangle + i\langle Y_i \rangle . ]
Local phase order is reported only when the total transverse visibility is non-zero:
[ V = \frac{1}{N}\sum_i |z_i|, \qquad R_V = \frac{|\sum_i z_i|}{\sum_i |z_i|}. ]
When (\sum_i |z_i|\le 10^{-12}), phase_defined is false and the
compatibility value R is stored as zero. A zero value in that case means
“unobservable local phase”, not antiphase locking.
The separate pair diagnostic is
[ C_{XY} = \binom{N}{2}^{-1}\sum_{i<j} 2\left|\langle \sigma_i^+\sigma_j^-\rangle\right|, \qquad 0\le C_{XY}\le 1. ]
C_XY measures transverse exchange coherence. It is a custom finite-system
diagnostic, not a spontaneous-synchronisation certificate.
Each pure state (\rho) is compared with its computational-basis-dephased control
[ \mathcal D(\rho)=\sum_x |x\rangle\langle x|\rho|x\rangle\langle x|. ]
The pair has identical computational-basis populations and local (Z)-marginals. It is not matched on Hamiltonian energy or every correlation, so a difference identifies an off-diagonal-coherence contribution, not a unique causal contribution from entanglement. The separable product state is retained as an attribution control for precisely this reason.
Initial pure-state entanglement is described by the mean normalised one-qubit linear entropy,
[ \bar L = \frac{1}{N}\sum_i 2\left(1-\operatorname{Tr}\rho_i^2\right). ]
This value is used only for the pure initial states. It is not applied as an entanglement measure to the mixed dephased controls.
The committed evidence uses the Paper-27 four-qubit coupling matrix,
omega = [1.329, 2.61, 0.844, 1.52], (t\in[0,2]), and 20 exact time steps.
The table reports time-averaged (C_{XY}).
| Initial state | Initial (\bar L) | Pure state | Dephased control | Difference |
|---|---|---|---|---|
| Product | 0 | 0.419481647 | 0.154436520 | 0.265045127 |
| Bell pairs | 1 | 0.132115673 | 0.091096078 | 0.041019595 |
| GHZ | 1 | 0 | 0 | 0 |
| W | 0.75 | 0.366405312 | approximately 0 | 0.366405312 |
Bell-pair and W states differ from their dephased controls. GHZ is the zero-difference negative control. The separable product state also differs from its control, and by more than the Bell-pair row. The evidence therefore classifies the result as an initial-coherence observation that is not entanglement-specific.
The digest-bound JSON and Markdown records are in
data/entanglement_sync_product/. Reproduce them with:
python scripts/run_entanglement_sync_evidence.pyfrom scpn_quantum_control.analysis.entanglement_enhanced_sync import (
compare_initial_states_with_dephased_controls,
)
comparisons = compare_initial_states_with_dephased_controls(
K,
omega,
t_max=2.0,
n_steps=20,
)simulate_sync_trajectory(...) and compare_all_initial_states(...) expose
both observables and the phase-defined flags. The legacy
entanglement_advantage(...) name remains for compatibility, but returns only
a descriptive comparison with a governed no-advantage certificate. It no
longer reports a convergence speedup.
- Fiderer, Kuś, and Braun formulate qubit phase synchronisation relative to a fixed basis (Phys. Rev. A 94, 032336).
- Galve, Giorgi, and Zambrini review why quantum synchronisation measures are model-dependent and can disagree (arXiv:1610.05060).
- Roulet and Bruder connect phase locking and entanglement in a driven-dissipative spin-1 model (Phys. Rev. Lett. 121, 063601). That result does not transfer automatically to this closed qubit model.
These sources motivate careful phase and correlation diagnostics; none
validates C_XY as a universal synchronisation measure or supports the former
lowered-critical-coupling claim.
The evidence is local deterministic simulation for one Hamiltonian and time grid. It does not establish an entanglement-specific cause, lower critical coupling, universal enhancement, quantum advantage, hardware fidelity, or control authority.