Born's rule emerges from finite-dimensional unitary evolution.
Paper: Where do the other worlds go? A toy-model test of Born-rule filtering in a local Ising chain — L. Eschenauer (2026)
This project demonstrates that a finite-dimensional quantum system, evolving under local unitary dynamics, preferentially retains Born-typical statistics — without invoking Born's rule as a postulate. This is a circuit-level implementation of the recoherence mechanism discovered by Strasberg, Schindler, Wang & Winter (2023–2026).
A 9-qubit system evolving under exact Ising dynamics shows that when the number of events exceeds what the finite-dimensional Hilbert space (D=512) can faithfully record, Born-typical frequency classes remain decoherent while combinatorial-peak frequencies become recoherent:
| ε (recoherence) | Interpretation | |
|---|---|---|
| Born frequency (n₁/L ≈ 0.75) | low | Decoherent — stable records |
| Combinatorial peak (n₁/L = 0.5) | ~3× higher | Recoherent — records lost |
| Gap | 3.1× (10-seed avg) |
This result is robust over 10 different initial states and matches Strasberg's random matrix model at the same Hilbert space dimension:
The effect emerges as the system is overloaded with more events than it can store:
In a finite-dimensional Hilbert space, there aren't enough orthogonal directions to keep all histories decoherent. The ones that survive are the Born-typical ones — histories whose frequencies deviate from Born's rule become strongly recoherent, losing their ability to leave detectable records. This reveals an interesting correlation between Born's rule and the structure of decoherence.
Whether this constitutes a full derivation of Born's rule remains an open problem (as Strasberg et al. discuss in Sec. V.B). This does not prove or disprove any interpretation of quantum mechanics. Both Copenhagen and Everettian physicists agree on the circuit's output. The result shows that Born-typical statistics are selected by the geometry of finite Hilbert spaces, even if the precise mechanism is not yet fully understood.
Strasberg et al. demonstrated Born-rule filtering using dense random matrix Hamiltonians (maximally non-local). Here we show a demonstration with local dynamics and exact matrix exponentiation — a transverse-field Ising model in the near-critical regime (H = J/2·ΣZᵢZᵢ₊₁ + hx·ΣXᵢ, integrable).
There is no separate recorder register. The 9-qubit system IS the finite-dimensional recorder, exactly as in Strasberg's formalism. The "outcome" at each step is defined by the Hamming weight of the qubits (≥ 4 → outcome "1", giving p₁ ≈ 0.746). This is computed analytically in the frequency-class decomposition, not by a physical gate.
The evolution is just L applications of the exact Ising unitary U = exp(-iHΔt):
|ψ₀⟩ ─── [Ising U₁] ─── [Ising U₂] ─── ... ─── [Ising U_L] ─── Measure
Each Ising layer consists of nearest-neighbor ZZ couplings + single-qubit Rx and Rz rotations. All gates are deterministic. No randomness, no mid-circuit measurements.
poetry install
poetry run pytest tests/ -v # 15 tests
poetry run python scripts/simulate.py # full results (~5 min)src/recohere/
analysis.py — Gram matrix and recoherence parameter ε (Strasberg eq. 6)
ising_direct.py — Ising evolution with analytical coarse-graining (the result)
strasberg.py — Strasberg's random matrix model (positive control)
branches.py — Branch-level analysis: tracks all 2^L individual histories
scripts/
simulate.py — Frequency-class simulation: multi-seed Ising + Strasberg + plots
branches.py — Branch-level analysis: death of worlds, census, Gram heatmap
one_reality.py — Extract and visualize individual branches (one "reality")
robustness.py — Robustness checks across thresholds and time steps
results/
overlay.png — Strasberg vs Ising, averaged over 10 seeds with ±1σ bands
robustness.png — All 10 individual seed runs showing consistency
emergence.png — ε panels as L grows from 2 to 15
tracking.png — ε_born vs ε_comb divergence as L increases
branch_*.png — Branch-level: death, census, Gram heatmap, survivors
one_reality.png — Individual branch visualization
GitHub provides a "Cite this repository" button in the sidebar (via CITATION.cff).
@misc{Eschenauer2026,
author = {Eschenauer, Laurent},
title = {Where do the other worlds go? A toy-model test of Born-rule filtering in a local Ising chain},
year = {2026},
url = {https://github.com/eschnou/quantum-recoherence}
}- Strasberg et al., "Approximate Decoherence, Recoherence and Records in Isolated Quantum Systems", arXiv:2601.19703 (2026)
- Strasberg & Schindler, "Shearing Off the Tree: Emerging Branch Structure and Born's Rule in an Equilibrated Multiverse", arXiv:2310.06755 (2023/2026)
- Strasberg, Reinhard & Schindler, "First Principles Numerical Demonstration of Emergent Decoherent Histories", Phys. Rev. X 14, 041027 (2024)