Entangled initial-state coherence study

BL-79 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.

Why the observable changed

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.

Matched controls and attribution

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.

Frozen evidence

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.py

API

from 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 BL-65 no-advantage certificate. It no longer reports a convergence speedup.

Literature boundary

• 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.

Claim boundary

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.