As a minimal feasibility check, a toy numerical experiment was implemented based on the CCF-inspired model \Gamma_{\text{eff}}(L) = \Gamma_{\text{env}}[1 + \chi (L/L_I)], with excess decoherence \Delta\Gamma(L) = \Gamma_{\text{eff}}(L) - \Gamma_{\text{env}} = \chi,\Gamma_{\text{env}}(L/L_I) at fixed hardware and environment. Synthetic datasets are generated by choosing a set of coherence-load values L (e.g., a uniform grid from L = 1 to L = 20), computing \Gamma_{\text{eff,true}}(L) = \Gamma_{\text{env}}[1 + \chi_{\text{true}} (L/L_I)], and adding Gaussian noise with tunable standard deviation to mimic experimental uncertainty. A simple linear regression \Gamma_{\text{eff}}(L) \approx a + bL is then used to estimate a slope b and recover \chi_{\text{est}} = bL_I/\Gamma_{\text{env}}, along with a p‑value for “slope ≠0.” In this toy setting, low-to-moderate noise combined with a broad, dense range of coherence loads yields \chi_{\text{est}} close to the injected \chi_{\text{true}} with very small p‑values, while higher noise and narrow L-ranges produce fits statistically consistent with χ = 0, illustrating how the CCF excess-decoherence slope can be either clearly observable or effectively hidden, depending on experimental resolution and accessible coherence complexity.
import numpy as np
import matplotlib.pyplot as plt
from scipy import stats
# CCF-inspired model:
# Gamma_eff(L) = Gamma_env * [1 + chi * (L / L_I)]
# DeltaGamma(L) = Gamma_eff(L) - Gamma_env = chi * Gamma_env * (L / L_I)
Gamma_env = 1.0 # baseline decoherence rate
chi_true = 0.3 # injected coherence-capacity parameter
L_I = 5.0 # reference complexity scale
L_values = np.linspace(1, 20, 20) # coherence-load values
np.random.seed(0)
noise_sigma = 0.15 # noise level; adjust to explore detectability
Gamma_eff_true = Gamma_env * (1 + chi_true * (L_values / L_I))
Gamma_eff_obs = Gamma_eff_true + np.random.normal(0, noise_sigma, size=L_values.shape)
slope, intercept, r_value, p_value, std_err = stats.linregress(L_values, Gamma_eff_obs)
chi_est = slope * L_I / Gamma_env
print(f"True chi: {chi_true:.3f}")
print(f"Estimated chi: {chi_est:.3f}")
print(f"p-value for slope != 0: {p_value:.3e}")
L_fit = np.linspace(L_values.min(), L_values.max(), 200)
Gamma_fit = intercept + slope * L_fit
plt.figure(figsize=(6,4))
plt.scatter(L_values, Gamma_eff_obs, label='Simulated data')
plt.plot(L_fit, Gamma_fit, 'r-', label='Linear fit')
plt.xlabel("Coherence load L")
plt.ylabel("Effective decoherence rate Γ_eff(L)")
plt.legend()
plt.grid(True)
plt.show()