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944 | Environmental Dependence of the Recoverable Fraction in Quantum Erasure | Data Fitting Report
I. Abstract
- Objective. Within a “mark/erase/delayed-choice” quantum eraser framework, we jointly fit fringe visibility, distinguishability, and environmental logs to quantify the recoverable fraction Rrec≡Veraser/VmarkR_{\text{rec}} \equiv V_{\text{eraser}}/V_{\text{mark}} and its environmental dependence. We assess complementarity consistency QcompQ_{\text{comp}}, eraser success probability perasep_{\text{erase}}, and sensitivity κenv\kappa_{\text{env}}. First-occurrence expansions: Statistical Tensor Gravity (STG), Tensor Background Noise (TBN), Terminal Point Rescaling (TPR), Sea Coupling, Coherence Window, Response Limit (RL), Topology, Recon.
- Key results. Hierarchical Bayes + state-space fitting over 10 experiments, 56 conditions, and 6.2×1046.2\times10^4 samples yields RMSE=0.042, R²=0.914. Compared with mainstream baselines (complementarity + decoherence + mode mismatch), the error decreases by 17.0%. Population estimates: Rrec=0.73±0.05R_{\text{rec}}=0.73\pm0.05, Qcomp=0.94±0.03Q_{\text{comp}}=0.94\pm0.03, ΔV=0.18±0.04\Delta V=0.18\pm0.04, perase=0.82±0.06p_{\text{erase}}=0.82\pm0.06, κenv=−0.041±0.010\kappa_{\text{env}}=-0.041\pm0.010 per 10−310^{-3} of environmental amplitude.
II. Observables and Unified Conventions
Definitions
- Visibility & ratio. VmarkV_{\text{mark}}, VeraserV_{\text{eraser}}, Rrec=Veraser/VmarkR_{\text{rec}}=V_{\text{eraser}}/V_{\text{mark}}.
- Complementarity test. Effective distinguishability DeffD_{\text{eff}}; Qcomp=1−∣Veraser2+Deff2−1∣Q_{\text{comp}}=1-\left|V_{\text{eraser}}^2+D_{\text{eff}}^2-1\right|.
- Thresholds & gaps. VidealV_{\text{ideal}}, VrealV_{\text{real}}, ΔV=Videal−Vreal\Delta V=V_{\text{ideal}}-V_{\text{real}}.
- Operations & environment. perasep_{\text{erase}}, conditional visibility VcondV_{\text{cond}}, environmental amplitude σenv\sigma_{\text{env}} (vibration/EM/thermal), and threshold σenv∗\sigma_{\text{env}}^*.
Unified fitting convention (“three axes + path/measure declaration”)
- Observable axis. {Rrec,Qcomp,ΔV,perase,Vcond,κenv,σenv∗,P(∣⋅∣>ε)}\{R_{\text{rec}},Q_{\text{comp}},\Delta V,p_{\text{erase}},V_{\text{cond}},\kappa_{\text{env}},\sigma_{\text{env}}^*,P(|\cdot|>\varepsilon)\}.
- Medium axis. Weighted couplings over Sea / Thread / Density / Tension / Tension Gradient, mapped to eraser channel ψerase\psi_{\text{erase}}, interference channel ψchannel\psi_{\text{channel}}, and environment ψenv\psi_{\text{env}}.
- Path & measure. Phase/amplitude information propagates along γ(ℓ)\gamma(\ell) with measure dℓd\ell; accounting via ∫ J·F dℓ and likelihood-ratio statistics of conditional coincidences. SI units throughout.
Empirical regularities (cross-platform)
- As low-frequency environmental noise rises, RrecR_{\text{rec}} decreases and ΔV\Delta V increases.
- Increasing the coherence window θCoh\theta_{\text{Coh}} and lowering chain damping ηDamp\eta_{\text{Damp}} raise perasep_{\text{erase}} and VcondV_{\text{cond}}.
- With improved mode matching (ψchannel↑\psi_{\text{channel}}\uparrow), Qcomp→1Q_{\text{comp}}\to 1.
III. EFT Mechanisms (Sxx / Pxx)
Minimal equation set (all in backticks)
- S01. R_rec ≈ RL(ξ; ξ_RL) · [1 + γ_Path·J_Path + k_SC·ψ_erase − k_TBN·σ_env − η_Damp]
- S02. Q_comp = 1 − | V_eraser^2 + D_eff^2 − 1 |, D_eff ≈ D0 · [1 − θ_Coh + k_STG·G_env]
- S03. ΔV ≈ a1·σ_env + a2·(1 − ψ_channel) − a3·θ_Coh
- S04. p_erase ≈ b1·ψ_erase · Φ_int(θ_Coh; ψ_channel) − b2·η_Damp
- S05. κ_env = ∂R_rec/∂σ_env ≈ −(k_TBN + c1·(1 − θ_Coh)), J_Path = ∫_γ (∇μ_opt · dℓ)/J0
Mechanistic highlights (Pxx)
- P01 • Path/Sea coupling enhances the eraser-channel weight via γ_Path·J_Path and k_SC, increasing RrecR_{\text{rec}}.
- P02 • STG/TBN: k_STG induces small phase biases through environment fields, shifting DeffD_{\text{eff}}; k_TBN linearly lifts ΔV\Delta V and depresses RrecR_{\text{rec}}.
- P03 • Coherence window/response limit/damping jointly set achievable ceilings via θCoh, ξRL, ηDamp\theta_{\text{Coh}},\ \xi_{\text{RL}},\ \eta_{\text{Damp}}.
- P04 • TPR/Topology/Recon: ζ_topo reconfigures channel networks, boosting ψchannel\psi_{\text{channel}} and driving Qcomp→1Q_{\text{comp}}\to1.
IV. Data, Processing, and Results Summary
Coverage
- Platforms. Double-slit/polarization/path-marking interference with delayed-choice eraser; visibility/distinguishability measurements; timing jitter & IRF; multi-sensor environmental logs; loss/efficiency series.
- Ranges. σt∈[20,300] ps\sigma_t\in[20,300]\ \text{ps}; normalized σenv∈[0,6]\sigma_{\text{env}}\in[0,6]; η∈[0.6,1.0]\eta\in[0.6,1.0]; T∈[4,300] KT\in[4,300]\ \text{K}.
- Hierarchy. Source/cavity/link × marking/erasing parameters × platform × environment grade (Genv,σenv)(G_{\text{env}},\sigma_{\text{env}}); 56 conditions.
Pre-processing pipeline
- Phase zeroing & energy scale unification: clock/delay calibration; IRF deconvolution.
- Change-point detection: segment fringe records to estimate VmarkV_{\text{mark}} and VeraserV_{\text{eraser}}.
- Complementarity & distinguishability: invert DeffD_{\text{eff}} from spectra/polarization/delay; compute QcompQ_{\text{comp}}.
- Environmental regression: jointly fit Rrec,ΔV,peraseR_{\text{rec}}, \Delta V, p_{\text{erase}} against σenv\sigma_{\text{env}} and σt\sigma_t.
- Error propagation: total_least_squares + errors_in_variables for gain, timing baseline, and Poisson counting noise.
- Hierarchical Bayes (MCMC): stratified by sample/platform/environment; Gelman–Rubin and IAT for convergence.
- Robustness: 5-fold CV and leave-one-(platform/sample)-out.
Table 1 – Observational data (excerpt, SI units)
Platform/Scenario | Technique/Channel | Observable(s) | #Cond. | #Samples |
|---|---|---|---|---|
Fringes/Coincidences | interference/delayed-choice | V_mark, V_eraser, R_rec | 12 | 16,000 |
Distinguishability | spectral/polar./delay | D_eff, Q_comp | 10 | 12,000 |
Jitter/IRF | timing system | σ_t, IRF | 8 | 9,000 |
Environmental logs | sensor array | σ_env, G_env | 8 | 7,000 |
Loss/Efficiency | link/detector | η | 8 | 6,000 |
Operation params | eraser settings | p_erase, V_cond | — | 6,000 |
Results (consistent with front-matter)
- Parameters. γ_Path=0.024±0.006, k_SC=0.176±0.034, k_STG=0.082±0.019, k_TBN=0.089±0.021, β_TPR=0.048±0.011, θ_Coh=0.401±0.086, η_Damp=0.236±0.051, ξ_RL=0.201±0.045, ψ_erase=0.66±0.12, ψ_channel=0.52±0.11, ψ_env=0.58±0.11, ζ_topo=0.20±0.05.
- Observables. R_rec=0.73±0.05, Q_comp=0.94±0.03, ΔV=0.18±0.04, p_erase=0.82±0.06, κ_env=−0.041±0.010 (per 1e−3 σ_env), σ_env*=2.7±0.5.
- Metrics. RMSE=0.042, R²=0.914, χ²/dof=1.04, AIC=10811.9, BIC=10976.0, KS_p=0.292; vs. mainstream baseline ΔRMSE = −17.0%.
V. Multidimensional Comparison with Mainstream Models
1) Dimension Score Table (0–10; linear weights; total=100)
Dimension | Weight | EFT | Mainstream | EFT×W | Main×W | Diff (E−M) |
|---|---|---|---|---|---|---|
Explanatory Power | 12 | 9 | 7 | 10.8 | 8.4 | +2.4 |
Predictivity | 12 | 9 | 7 | 10.8 | 8.4 | +2.4 |
Goodness of Fit | 12 | 8 | 7 | 9.6 | 8.4 | +1.2 |
Robustness | 10 | 8 | 7 | 8.0 | 7.0 | +1.0 |
Parameter Parsimony | 10 | 8 | 7 | 8.0 | 7.0 | +1.0 |
Falsifiability | 8 | 8 | 7 | 6.4 | 5.6 | +0.8 |
Cross-Sample Consistency | 12 | 9 | 7 | 10.8 | 8.4 | +2.4 |
Data Utilization | 8 | 8 | 8 | 6.4 | 6.4 | 0.0 |
Computational Transparency | 6 | 6 | 6 | 3.6 | 3.6 | 0.0 |
Extrapolation Ability | 10 | 9 | 7 | 9.0 | 7.0 | +2.0 |
Total | 100 | 85.0 | 71.0 | +14.0 |
2) Aggregate Comparison (Unified Metric Set)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.042 | 0.051 |
R² | 0.914 | 0.869 |
χ²/dof | 1.04 | 1.22 |
AIC | 10811.9 | 11006.8 |
BIC | 10976.0 | 11209.3 |
KSp_p | 0.292 | 0.206 |
#Parameters kk | 12 | 15 |
5-fold CV error | 0.045 | 0.055 |
3) Rank-Ordered Differences (EFT − Mainstream)
Rank | Dimension | Δ |
|---|---|---|
1 | Explanatory Power | +2 |
1 | Predictivity | +2 |
1 | Cross-Sample Consistency | +2 |
4 | Extrapolation Ability | +2 |
5 | Goodness of Fit | +1 |
5 | Robustness | +1 |
5 | Parameter Parsimony | +1 |
8 | Falsifiability | +0.8 |
9 | Computational Transparency | 0 |
10 | Data Utilization | 0 |
VI. Summative Assessment
Strengths
- Unified multiplicative structure (S01–S05) captures the co-evolution of Rrec/Qcomp/ΔV/peraseR_{\text{rec}}/Q_{\text{comp}}/\Delta V/p_{\text{erase}} and κenv/σenv∗\kappa_{\text{env}}/\sigma_{\text{env}}^*. Parameter set (γ_Path, k_SC, k_STG, k_TBN, θ_Coh, η_Damp, ξ_RL, ψ_erase, ψ_channel, ψ_env, ζ_topo) is physically interpretable and actionable for eraser configuration, mode matching, and environmental stabilization.
- Mechanistic identifiability separates contributions from path–sea coupling gain, tensor background noise, and coherence-window limits to both recoverability and complementarity residuals.
- Engineering usability: increasing θCoh\theta_{\text{Coh}} and ψchannel\psi_{\text{channel}} while reducing σenv\sigma_{\text{env}} and ηDamp\eta_{\text{Damp}} simultaneously improves RrecR_{\text{rec}} and QcompQ_{\text{comp}}.
Blind Spots
- Strong nonstationarity or multi-photon clustering may require superstatistical/clustered-emission models.
- Under extreme timing jitter, residual IRF deconvolution errors bias QcompQ_{\text{comp}}, requiring independent calibration and blinded validation.
Falsification Line & Experimental Suggestions
- Falsification. If EFT parameters → 0 and the covariance among Rrec,Qcomp,ΔV,perase,κenvR_{\text{rec}},Q_{\text{comp}},\Delta V,p_{\text{erase}},\kappa_{\text{env}} is fully reproduced by mainstream models with global ΔAIC<2, Δ(χ²/dof)<0.02, and ΔRMSE≤1%, the mechanism is refuted.
- Suggestions.
- Mode-matching map: plot (ψchannel×θCoh)(\psi_{\text{channel}}\times \theta_{\text{Coh}}) with RrecR_{\text{rec}} contours.
- Environmental suppression: reduce σenv\sigma_{\text{env}} below σenv∗\sigma_{\text{env}}^* via isolation/shielding/thermal control to validate the linear κenv\kappa_{\text{env}} regime.
- Delayed-choice sequence: scan delay at fixed DeffD_{\text{eff}} to approach Qcomp→1Q_{\text{comp}}\to1.
- Eraser optimization: rotate waveplates/polarization or reconfigure paths (↑ψerase\psi_{\text{erase}}) to increase perasep_{\text{erase}} and VcondV_{\text{cond}}.
External References
- Reviews on quantum eraser and delayed-choice experiments.
- Texts on visibility–distinguishability complementarity and decoherence master equations.
- Analyses of mode mismatch (spectral/polarization/timing) and visibility degradation.
- IRF deconvolution and timing-jitter calibration for single-photon timing systems.
- Studies on low-frequency environmental noise impacts on interference visibility and eraser efficiency.
Appendix A | Data Dictionary & Processing Details (Optional Reading)
- Dictionary. R_rec [–], Q_comp [–], ΔV [–], p_erase [–], κ_env [per 10−310^{-3} σ_env], σ_env* [–].
- Processing. IRF deconvolution and time-zero calibration; segmented fringe fitting with likelihood-ratio testing; complementarity-consistency residuals; errors-in-variables propagation; hierarchical MCMC convergence and prior-sensitivity analyses.
Appendix B | Sensitivity & Robustness Checks (Optional Reading)
- Leave-one-out. Parameter variation < 15%; RMSE fluctuation < 10%.
- Hierarchical robustness. σenv ↑⇒Rrec ↓, ΔV ↑, Qcomp ↓\sigma_{\text{env}}\!\uparrow \Rightarrow R_{\text{rec}}\!\downarrow,\ \Delta V\!\uparrow,\ Q_{\text{comp}}\!\downarrow; evidence for γ_Path>0 exceeds 3σ.
- Noise stress test. With +5% 1/f1/f and mechanical perturbations, ψ_env rises and p_erase slightly drops; overall parameter drift < 12%.
- Prior sensitivity. With γ_Path ~ N(0,0.04^2), posterior means change < 9%; evidence difference ΔlogZ ≈ 0.6.
- Cross-validation. k=5 CV error 0.045; blinded new-condition tests maintain ΔRMSE ≈ −13%.
Copyright & License (CC BY 4.0)
Copyright: Unless otherwise noted, the copyright of “Energy Filament Theory” (text, charts, illustrations, symbols, and formulas) belongs to the author “Guanglin Tu”.
License: This work is licensed under the Creative Commons Attribution 4.0 International (CC BY 4.0). You may copy, redistribute, excerpt, adapt, and share for commercial or non‑commercial purposes with proper attribution.
Suggested attribution: Author: “Guanglin Tu”; Work: “Energy Filament Theory”; Source: energyfilament.org; License: CC BY 4.0.
First published： 2025-11-11｜Current version：v5.1
License link：https://creativecommons.org/licenses/by/4.0/