Trapped-Ion Page-Wootters Experiment: Protocol for Testing Ultrametricity

In the paper's words

We propose a trapped-ion experiment to test the Sufficient Condition Theorem: that diagonal clock-rest coupling produces exact ultrametric conditional state overlaps (UVR = 0%), while nondiagonal coupling produces a universal ~32% violation rate. All required capabilities are established with current trapped-ion technology. Estimated timeline: 8 weeks on an existing apparatus. --- Published under QNFO-ULA. legal.qnfo.org

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

Numbers it reports

1. For $M = 4$ motional Fock states (truncated): Apply displacement $D(\alpha)$ with varying $\alpha$ Measure parity $\hat{P} = e^{i\pi\hat{a}^\dagger\hat{a}}$ of the motional state Reconstruct Wigner function $W(\alpha)$ Extract density matrix $\rho_k$ via inverse Radon transform Measurement time: $\sim$1 ms per shot $\times$ 10,000 shots $\times$ 4 clock states = 40 seconds per data point.

   from 3.3 Motional State Tomography

2. Error Source Magnitude Effect on UVR Tomography reconstruction noise $\Delta O_{ij} \approx 0.02$–$0.05$ Falsely elevates UVR by $\sim$3–8% Incomplete adiabatic preparation Fidelity $\lt 0.99$ Mixes ideal WDW state with excited states Motional heating ($\dot{\bar{n}}$) $\sim$10 quanta/s Decoherence during tomography Magnetic field fluctuations $\Delta B/B \sim 10^{-4}$ Dephasing of clock states Laser intensity noise $\Delta\Omega/\Omega \sim 10^{-3}$ Fluctuations in $\hat{H}_{CR}$

   from 6.1 Sources of Systematic Error

3. We propose a trapped-ion experiment to test the Sufficient Condition Theorem: that diagonal clock-rest coupling $\hat{H}_{CR}$ in the clock eigenbasis produces exact ultrametric conditional state overlaps (UVR = 0%), while nondiagonal coupling produces a universal ~32% violation rate.

   from Abstract

4. Adiabatic condition: Ramp time $\tau \gg \max_{n\neq 0} \hbar/|E_n - E_0| \approx 2\pi/\min(\Omega, \omega_z)$ For $\Omega/2\pi = 50$ kHz: $\tau \gtrsim 20$ $\mu$s (easily achievable; ion coherence times are $\sim$100 ms)

   from 2.2 Preparation Protocol

5. Regime $\hat{H}_{CR}$ Type Predicted UVR (theory) Predicted UVR (exp., $\epsilon=0.03$) Diagonal Carrier only 0.00% $\lt 5\%$ (limited by tomography noise) Nondiagonal Sideband coupling 32 ± 3% $28$–$37\%$ (statistical + systematic) Statistical significance: With $N = 4$ (4 triangles), the binomial probability of observing 0 violations in the diagonal regime when the true UVR is 32% is $p = (1-0.32)^4 \approx 0.21$ — insufficient.

   from 4.3 Predicted Outcomes

6. Requirement Value Status Single ion trapping (Yb$^+$ or Ca$^+$) Standard Paul trap ✅ Established [established] $N \geq 6$ resolvable Zeeman sublevels B-field $\approx$ 5–10 G ✅ Routine [established] Motional ground state cooling $\bar{n} \lt 0.1$ ✅ Routine (sideband cooling) [established] Individual addressing of transitions Multiple AOM channels ✅ Routine [established] Motional state tomography ($M \geq 4$) Displacement + parity ✅ Demonstrated (Leibfried et al., 1996; Lv et al., 2018) [established] Adiabatic state preparation Ramped laser intensity ✅ Routine [established] Coherence time $\gt $ 100 ms Magnetic shielding ✅ Achievable [established]

   from 5.1 Required Capabilities

How it is done

1. Author: QNFO Research | Date: 2026-07-01

   from Trapped-Ion Page-Wootters Experiment: Protocol for Testing Ultrametricity

2. We propose a trapped-ion experiment to test the Sufficient Condition Theorem: that diagonal clock-rest coupling $\hat{H}_{CR}$ in the clock eigenbasis produces exact ultrametric conditional state overlaps (UVR = 0%), while nondiagonal coupling produces a universal ~32% violation rate.

   from Abstract

3. This is the experiment's central advantage.

   from 1.3 Clock-Rest Interaction Engineering

4. We use adiabatic state preparation : Initialize: Prepare the system in a product state $|e_0\rangle \otimes |0\rangle$ (ground clock state, motional ground state) Ramp coupling: Slowly turn on $\hat{H}_{CR}$ from 0 to target strength, keeping the total Hamiltonian at each step Verify: The final state is an eigenstate of $\hat{H}$ with eigenvalue $E_0$ Energy shift for zero eigenvalue.

   from 2.2 Preparation Protocol

What would falsify it

1. Conditional state tomography at multiple clock readings yields the overlap matrix $O_{ij}$, whose Parisi ultrametricity violation rate is the observable.

   from Abstract

2. The CMB null result is then interpreted as evidence that early-universe clock-rest coupling was diagonal — the "classical ideal clock" regime.

   from 8. Connection to CMB and Bruhat-Tits

Where the claim stops

1. With $N = 6$ clock states (20 triangles) and $R = 10^4$ repetitions: Per-triangle binomial error on UVR: $\sigma_{\text{UVR}} \approx \sqrt{p(1-p)/R_{\text{tri}}}$ For $p = 0.32$ and 333 effective trials per triangle: $\sigma_{\text{UVR}} \approx 0.026$ Total UVR uncertainty: $\approx 3\%$

   from 6.2 Statistical Error

2. Open question for future work: Can the trapped-ion platform simulate a p-adic clock spectrum and test the conjecture that ultrametricity is equivalent to the p-adic tree structure of Bruhat-Tits buildings?

   from 8. Connection to CMB and Bruhat-Tits

Where to read

1. Abstract91 words · 1 min
2. 1. Physical Implementation206 words · 1 min
3. 2. WDW State Preparation179 words · 1 min
4. 3. Conditional State Measurement150 words · 1 min
5. 4. Ultrametricity Test157 words · 1 min
6. 5. Experimental Requirements212 words · 1 min
7. 6. Noise Analysis and Error Budget139 words · 1 min
8. 7. Timeline and Resource Estimate64 words · 1 min
9. 8. Connection to CMB and Bruhat-Tits96 words · 1 min