Decoherence as the Ultimate Consequence of Linear Accumulation

In the paper's words

We argue that quantum decoherence is not a qualitatively distinct phenomenon requiring new physical postulates, but the inevitable long-time consequence of a single structural assumption: that small environmental perturbations accumulate linearly in the phase space of a system–environment composite. We formalize this claim with a discrete kick model in which each system–environment interaction deposits an independent phase perturbation of characteristic magnitude σ_φ; linear accumulation of perturbation variance yields the exponential coherence decay C(t)=exp(-Γ_kickσ_φ² t/2), with dephasing time T₂=2/(Γ_kickσ_φ²). Equivalently, in a multi-mode formulation, the total decoherence rate is the linear sum Γ_Σ=∑_k=1^NΓ_k over N environmental modes, so macroscopic classicality is a counting statement. [...]

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. This work confronts the High-Luminosity LHC upgrade, with a factor of $5$–$10$ increase in peak luminosity proportionally increasing occupancy and radiation damage of tracking sensors.

   from 2. Background and Related Work

2. Our Section 4.3 and 4.6 supply the quantitative baseline (Zeno crossover time, $1/N$ scaling ratio) that such a falsification must respect; if deterministic trigger semantics mattered, relaxation statistics would not depend solely on accumulated perturbation statistics, breaking our central claim. [11] QNFO: Monistic Reality | https://papers.qnfo.org/papers/converging-reality/.

   from 2. Background and Related Work

3. Our Section 4.5 makes the single-generation budget explicit and identifies the threshold condition as a condition on the per-interaction perturbation magnitude.

   from 2. Background and Related Work

4. If the kick rate is reduced tenfold to $\Gamma_{\mathrm{kick}}'=10^{3}\,\mathrm{s^{-1}}$ (the strategy exemplified by the trapping engineering of [3], [5]), then $$T_2'=\frac{2}{10^{3}\times10^{-6}}\,\mathrm{s}=2000\,\mathrm{s},$$ a tenfold improvement — decoherence time scales exactly inversely with kick rate, the same proportionality by which a $5$–$10\times$ luminosity increase in [6] proportionally shortens detector degradation times.

   from 4.2 Worked numerical example (fully explicit)

5. The ratio $$\frac{t_{\mathrm{cycle}}}{t_{\mathrm{Z}}}=\frac{2.5\times10^{-8}\,\mathrm{s}}{10^{-15}\,\mathrm{s}}=2.5\times10^{7}$$ shows that every operational timescale in the cited experimental contexts lies seven or more orders of magnitude inside the linear-accumulation regime.

   from 4.3 Zeno crossover time

6. The ratio of coherence times is $$\frac{T_2^{(1)}}{T_2^{(2)}}=\frac{10^{5}\,\mathrm{s}}{10\,\mathrm{s}}=10^{4}=\frac{N_2}{N_1}.$$ Step 5.

   from 4.4 Linear-in-$N$ scaling of $T_2$

How it is done

1. With fully shown arithmetic we compute: for $\sigma_\phi=10^{-3}\,\mathrm{rad}$ delivered at $\Gamma_{\mathrm{kick}}=10^4\,\mathrm{s^{-1}}$, $T_2=200\,\mathrm{s}$; for per-mode rate $\Gamma_k=10^{-10}\,\mathrm{s^{-1}}$, $T_2$ scales as $1/N$ from $10^5\,\mathrm{s}$ ($N=10^5$) to $10^{-13}\,\mathrm{s}$ ($N=10^{23}$); the Zeno-to-linear crossover sits at $t_{\mathrm{Z}}=\tau_B\sim10^{-15}\,\mathrm{s}$; and a circuit with $G=10^6$ gates at $p=10^{-10}$ has failure probability $9.995\times10^{-5}$, rising to near-certainty at $p=10^{-5}$.

   from Abstract

2. From this single structural assumption we derive exponential decoherence as a theorem with a computable rate.

   from 1. Introduction

3. The bibliography available for this preprint is drawn from adjacent fields rather than from the decoherence literature proper; we use each work for the structural feature it exemplifies, and we flag this limitation explicitly in Section 6. [1] The Araucaria Project: Improving the cosmic distance scale.

   from 2. Background and Related Work

4. Consider a two-level system with basis $\{|0\rangle,|1\rangle\}$ interacting sequentially with $N$ environmental degrees of freedom (modes), one per elementary interaction.

   from 3.1 The kick-accumulation model

What would falsify it

1. We state the conditions under which the thesis would be falsified, connect it to the pre-registered QNFO falsification program, and document explicitly where independent drafts of this work diverged and which convention was adopted.

   from Abstract

2. Second, it gives a sharp falsification criterion: if one could engineer an environment whose perturbations accumulate sub-linearly in variance, the resulting coherence time would exceed the linear-accumulation prediction in a quantifiable way.

   from 1. Introduction

3. Third, it connects to the QNFO program's pre-registered falsification of deterministic measurement-triggered relaxation [10]: our framework predicts that relaxation statistics depend only on the accumulated perturbation distribution, not on any trigger semantics of measurement — precisely the hypothesis under test in [10].

   from 1. Introduction

Where the claim stops

1. The model assumes independent, identically distributed, zero-mean kicks and a pure-dephasing, factorizable bath.

   from 6. Discussion

2. The small-kick expansion requires $\sigma_\phi^2/2\ll1$; for strong individual kicks the product formula must be evaluated exactly and the exponential law can fail at short times.

   from 6. Discussion

3. The derived law $C(t)=\exp(-\Gamma_{\mathrm{kick}}\sigma_\phi^2 t/2)$ with $T_2=2/(\Gamma_{\mathrm{kick}}\sigma_\phi^2)$, the exact inverse scaling of $T_2$ with kick rate (R4), the exact $1/N$ scaling with mode number (ratio $T_2^{(1)}/T_2^{(2)}=10^4=N_2/N_1$ for the stated inputs, R6), the Zeno crossover at $t_{\mathrm{Z}}=\tau_B=10^{-15}\,\mathrm{s}$ (R5), the $2000\times$ systematic-error penalty (R7), and the circuit-level failure probabilities $P_{\mathrm{fail}}\approx9.995\times10^{-5}$ for $G=10^6$, $p=10^{-10}$ and $P_{\mathrm{fail}}\approx0.99995$ for $p=10^{-5}$ (R8) are all computed with fully shown arithmetic from labeled assumptions.

   from 7. Conclusion

Where to read

1. Abstract186 words · 1 min
2. 1. Introduction355 words · 2 min
3. 2. Background and Related Work881 words · 4 min
4. 3. Methods410 words · 2 min
5. 4. Analysis782 words · 3 min
6. 5. Results215 words · 1 min
7. 6. Discussion329 words · 1 min
8. 7. Conclusion161 words · 1 min
9. Appendix A. Divergence report258 words · 1 min
10. Appendix B. Claim attribution362 words · 2 min