# Direct Observation of Discrete Decoherence Steps: Experimental Protocol


**2026年6月**

## ​‌‍​‌‍​‌‍​‌‍​‌‍​‌‍​‌‍​‌‍​‌‍​‌‍​‌‍​‌‍​‌‍​‌‍​‌‍​‌‍​‌‍​A Test of the Structural Decoherence Convergence Theorem

### Lin Xiaohei (林小黑) — June 21, 2026

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## ⚠️ Copyright Notice

**Designer: Lin Xiaohei (China).** This experimental protocol is a direct test of Predictions 1-3 from *The Structural Axiom System: Complete Mathematical Formalization* (June 21, 2026).

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## Abstract

We present a detailed experimental protocol to test the discrete decoherence step prediction of the Structural Axiom System. The theory predicts that quantum decoherence proceeds through a finite number of discrete jumps, not continuous exponential decay. The number of observable steps $k \approx \log_2 N$ where $N$ is the number of independent environmental degrees of freedom, and step heights are quantized as $\Delta\rho_j \propto \dim(\ker C_j)/\dim(\mathcal{H})$. We propose three experimental platforms — superconducting qubits, trapped ions, and nitrogen-vacancy centers — each with specific parameters optimized for observing 3-12 discrete steps. Required temporal resolution is sub-microsecond to capture step transitions within decoherence times of 1-100 μs.

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## 1. Theoretical Prediction

From the Decoherence Convergence Theorem (Theorem 3.1):

$$\frac{d\rho}{dt} = -\nabla_\rho \mathcal{F}[\rho] + i[\rho, H_{\text{eff}}]$$

The constraint manifold $\mathcal{M}$ admits a filtration $\mathcal{M}_0 \supset \mathcal{M}_1 \supset \cdots \supset \mathcal{M}_k$, and $\rho(t)$ undergoes discrete jumps at each boundary $\partial\mathcal{M}_j$.

**Prediction**: The off-diagonal elements $\rho_{01}(t)$ decay as a step function, not an exponential:

$$\rho_{01}(t) = \rho_{01}(0) \cdot \prod_{j=1}^{k} \Theta(t - t_j) \cdot (1 - \delta_j)$$

where $t_j$ are step times and $\delta_j$ are step heights.

**Key observables**:
- $k$: number of discrete steps ($\approx \log_2 N$)
- $\delta_j$: step heights (quantized as $\dim(\ker C_j) / \dim(\mathcal{H})$)
- $t_j$: step times (logarithmically spaced)

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## 2. Platform 1: Superconducting Qubits (SQUID)

### 2.1 System Configuration

**Qubit**: Transmon qubit with $T_1 = 50$ μs, $T_2^* = 30$ μs
**Environment**: Purcell filter + 50Ω transmission line (engineered $N = 256-1024$ modes)
**Readout**: Dispersive readout via Josephson parametric amplifier (JPA), SNR > 10 dB at 1 μs integration
**Temporal resolution**: 100 ns (limited by JPA bandwidth ~10 MHz)

### 2.2 Experimental Sequence

```
1. Initialize: |ψ⟩ = (|0⟩ + |1⟩)/√2 via π/2 pulse (X/2 gate, 20 ns)
2. Wait: variable delay τ ∈ [0, 100] μs, in 100 ns steps
3. Tomography: measure ⟨σ_x⟩, ⟨σ_y⟩, ⟨σ_z⟩ via state tomography
4. Repeat: 10^5 shots per τ value for statistics
5. Extract: ρ01(τ) = (⟨σ_x⟩ + i⟨σ_y⟩)/2
```

### 2.3 Expected Signal

For $N = 256$ environmental modes: $k \approx \log_2 256 = 8$ steps.
Step times (theoretical): $t_j \approx T_2^* \cdot \ln(N/(N-2^{j}))$, giving roughly:
- $t_1 \approx 0.1$ μs
- $t_2 \approx 0.3$ μs
- $t_3 \approx 0.8$ μs
- $t_4 \approx 2$ μs
- $t_5 \approx 5$ μs
- $t_6 \approx 12$ μs
- $t_7 \approx 25$ μs
- $t_8 \approx 50$ μs

With 100 ns resolution, steps 4-8 are resolvable (5 resolvable steps).

### 2.4 Distinguishing from Standard Decoherence

Standard prediction: $\rho_{01}(t) = \rho_{01}(0) \cdot e^{-t/T_2}$ — smooth exponential.
Structural prediction: $\rho_{01}(t)$ shows 5+ discrete drops at specific times.

**Statistical discrimination**: Fit both models. The discrete-step model has $2k$ parameters ($k$ times + $k$ heights). The exponential model has 1 parameter ($T_2$). Use AIC/BIC: the discrete model should win if $k \geq 2$ observable steps ($\Delta\text{AIC} > 10$).

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## 3. Platform 2: Trapped Ions

### 3.1 System Configuration

**Ion**: $^{40}\text{Ca}^+$ or $^{171}\text{Yb}^+$, optical qubit ($S_{1/2} \leftrightarrow D_{5/2}$)
**Trap**: Linear Paul trap, secular frequency 1 MHz
**Cooling**: Doppler cooling to $\bar{n} \approx 5$ motional quanta
**Environment**: Motional mode bath ($N \approx 50-200$ phonon modes)
**Readout**: Electron shelving + fluorescence detection
**Temporal resolution**: 10 μs (limited by optical pumping time)

### 3.2 Expected Signal

For $N = 100$ motional modes: $k \approx 7$ steps.
Step times: $t_j \approx T_2 \cdot \ln(100/(100-2^j))$.
With $T_2 \approx 10$ ms (for optical qubits):
- $t_1 \approx 0.1$ ms
- $t_6 \approx 5$ ms
- $t_7 \approx 20$ ms

With 10 μs resolution, all 7 steps are resolvable in the motional decoherence channel.

### 3.3 Key Advantage

Trapped ions provide the cleanest separation between environmental modes (individual motional sidebands). Each mode can be independently characterized and its contribution to the $j$-th constraint layer identified. This enables *direct verification of step-height quantization*: measure $\delta_j$ for each step and check that $\delta_j / \delta_{j+1}$ equals a small integer ratio.

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## 4. Platform 3: Nitrogen-Vacancy Centers

### 4.1 System Configuration

**Defect**: Single NV$^-$ center in diamond, electronic spin qubit
**Environment**: $^{13}\text{C}$ nuclear spin bath ($N \approx 10^2-10^4$, controllable via isotope enrichment)
**Control**: Microwave π-pulses + dynamical decoupling (CPMG, XY8)
**Readout**: Spin-dependent fluorescence
**Temporal resolution**: 50 ns (limited by MW pulse duration)

### 4.2 Dynamic Decoupling Enhancement

Apply CPMG-$n$ sequence to slow down effective decoherence, stretching step times into the resolvable regime:

$$T_2^{\text{CPMG}} \approx T_2 \cdot n^{2/3}$$

For $n = 64$ pulses and $T_2^* = 2$ μs: $T_2^{\text{CPMG}} \approx 50$ μs.
Step times are stretched proportionally, making all $k \approx 8-12$ steps resolvable at 50 ns resolution.

### 4.3 Isotope Engineering

Use diamond samples with controlled $^{13}\text{C}$ concentration (0.01% to 1.1%) to systematically vary $N$ from $10^2$ to $10^4$. Verify the scaling $k \propto \log_2 N$.

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## 5. Data Analysis Protocol

### 5.1 Step Detection Algorithm

```
Input: ρ01(τ_i) for i = 1..M time points
Output: step times {t_j} and heights {δ_j}

1. Compute discrete derivative: Δρ[i] = |ρ01(τ_i) - ρ01(τ_{i-1})|
2. Identify peaks in |Δρ| exceeding 3σ above noise floor
3. Cluster proximate peaks (within 2× temporal resolution)
4. For each cluster, fit a sigmoid: ρ(t) = A + B/(1 + e^{-(t-t_0)/w})
5. Extract t_j = t_0, δ_j = B
6. Count k = number of detected steps
```

### 5.2 Statistical Hypothesis Test

**Null hypothesis $H_0$**: $\rho_{01}(t) = \rho_0 e^{-t/T_2}$ (continuous exponential)
**Alternative $H_1$**: $\rho_{01}(t) = \rho_0 \prod_j \Theta(t-t_j)(1-\delta_j)$ (discrete steps)

Test statistic: $F = \frac{(\text{RSS}_0 - \text{RSS}_1)/(2k-1)}{\text{RSS}_1/(M-2k)}$ (F-test for nested models).
Reject $H_0$ if $F > F_{\text{crit}}(2k-1, M-2k, \alpha=0.01)$.

### 5.3 Step-Height Quantization Test

If steps are detected, compute ratios $r_j = \delta_j / \delta_{j+1}$. Under the quantization hypothesis, $r_j$ should cluster near small integer ratios (1:1, 1:2, 2:1, 2:3, etc.). Test against a uniform distribution on $[0, 5]$ using a Kolmogorov-Smirnov test.

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## 6. Required Resources

| Item | Specification | Estimated Cost |
|------|--------------|:---:|
| Dilution refrigerator | Base temp < 20 mK | Existing |
| Superconducting qubit chip | 2-4 transmon qubits | Existing |
| JPA/parametric amplifier | Quantum-limited, BW > 10 MHz | Existing |
| AWG | 1 GSa/s, 4 channels | Existing |
| FPGA-based readout | Latency < 500 ns | Existing |
| Measurement time | ~72 hours (10^5 shots × 1000 τ values × 0.1 ms/cycle) | Existing |

**Total additional cost**: $0 (uses existing quantum computing testbed infrastructure).

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## 7. Collaboration Invitation

This protocol is designed for immediate implementation on existing quantum computing platforms. No new hardware is required. Any group with access to a superconducting qubit testbed, ion trap, or NV-center setup can perform the experiment within one week of setup time.

Groups with existing infrastructure:
- TU Delft (DiCarlo group) — superconducting qubits
- ETH Zurich (Wallraff group) — superconducting + NV
- Innsbruck (Blatt group) — trapped ions
- MIT (Englund group) — NV centers
- USTC (Pan group) — superconducting + ions

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## §7.1 An Unresolved Open Question

If discrete steps are observed, the next question is: can the step times be *predicted in advance* from independent characterization of the environmental mode spectrum? Or are they irreducibly emergent features of the coupling dynamics? The former would elevate the structural theory from post-diction to pre-diction. The author's position is not disclosed here.

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*Lin Xiaohei, June 21, 2026*
*Protocol design by Hermes Agent (则弟) under Lin Xiaohei's direction.*

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© 2026 林小黑 (Lin Xiaohei). All rights reserved.
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