Simulations
The Quantum Family Tree Theory makes a testable prediction: mutual information between particles should decay monotonically with genealogical distance, regardless of the specific quantum dynamics at each splitting event. The simulations below test this prediction directly.
All source code is available on GitHub.
Method
We construct quantum family trees at depths 2 through 4 using pure statevector simulation with NumPy. Each tree begins with a Bell pair. Each generation applies CNOT gates (parent to both children), a Hadamard gate (decoherence), and random unitary operators drawn from the Haar measure (quantum dynamics). After construction, ancestor qubits are traced out, leaving only leaf-state density matrices.
Mutual information I(A;B) = S(A) + S(B) − S(A,B) is computed between all leaf pairs, grouped by genealogical distance. Von Neumann entropy uses log base 2 (bits). For a pure Bell pair, MI = 2.0 bits; the observed sibling MI of 1.0 bits reflects genuine reduction from splitting and decoherence, not a normalization artifact.
System sizes
| Depth | Leaves | Qubits | Memory | Trees | Time/Tree |
|---|---|---|---|---|---|
| 2 | 4 | 6 | <1 MB | 200 | <1s |
| 3 | 8 | 14 | <1 MB | 100 | ~2s |
| 4 | 16 | 30 | 17 GB | 10 | ~9 hrs |
Depth 4 simulations ran on a Google Cloud n1-highmem-16 VM (104 GB RAM) over approximately 90 hours total. Each of the 10 trees used different Haar-random unitaries at every splitting event.
Results: Depth 4
Key finding
d = 0.854 × ln(1/MI) + 1.88 | R² = 0.993
Distance scales logarithmically with inverse entanglement. This is the holographic relationship. The Ryu-Takayanagi formula (S = A/4G) connects entanglement entropy to geometric area — our result demonstrates this connection emerging naturally from a quantum genealogical structure.
| ng | Mean MI | Std Dev | Classical | Relationship |
|---|---|---|---|---|
| 2 | 1.000 | 0.000 | 1.000 | Siblings |
| 4 | 0.076 | 0.048 | 0.500 | Cousins |
| 6 | 0.006 | 0.010 | 0.250 | 2nd Cousins |
| 8 | 0.001 | 0.002 | 0.125 | 3rd Cousins |
Where ng is the genealogical distance (total hops through the tree via the lowest common ancestor). The "Classical" column shows the naive kinship prediction (1/2)k−1 where k = ng/2. Observed decay is steeper by a factor of approximately 1.68, following MI = (1/2)1.678·ng.
The holographic fit
| Functional Form | Best Fit | R² |
|---|---|---|
| d = a ln(1/MI) + b | a=0.854, b=1.88 | 0.993 |
| MI = (1/2)α ng | α = 1.678 | 0.993 |
| MI = a · d−p | p = 6.25 | 0.999 |
The power-law fit achieves marginally higher R² on these four data points, but we prefer the logarithmic form on theoretical grounds: it maps directly to the Ryu-Takayanagi formula connecting entanglement entropy to geometric area. The power law may reflect effective small-N behavior that converges to logarithmic scaling at larger tree depths.
Ensemble universality
Each tree in the ensemble uses different random unitary operators at every splitting event, drawn from the Haar measure. Yet kinship decay emerges consistently: monotonic decay was observed in 80% of depth 4 trials, 82% at depth 3, and 100% at depth 2.
This universality is the strongest result. It demonstrates that the pattern is structural — arising from tree topology itself — rather than dependent on specific quantum dynamics. Whatever the unitaries are, whatever physics governs each splitting event, the genealogical structure alone is sufficient to produce emergent geometry.
| Depth | Leaves | Qubits | Monotonic % | Trees |
|---|---|---|---|---|
| 2 | 4 | 6 | 100% | 200 |
| 3 | 8 | 14 | 82% | 100 |
| 4 | 16 | 30 | 80% | 10 |
The role of decoherence
An important control result: without random unitaries, the quantum tree produces uniform mutual information between all leaf pairs regardless of kinship distance. The genealogical structure is encoded in the state but hidden by symmetry. Random unitaries — modeling decoherence — break this symmetry and expose the genealogical hierarchy.
Decoherence is therefore not merely why space "feels classical." It is the mechanism that makes kinship distance visible as geometric distance. Without decoherence, the family tree exists but cannot be read. This may be the conjecture's most surprising finding.
Next steps
Current limits: 30 qubits (16 leaves) is a toy universe. To observe true geometric emergence — curvature, topology, dimensional structure — we need hundreds of leaves. The path forward involves GPU-accelerated simulation (cuStateVec) and tensor network approximations (Matrix Product States), which could enable depths of 20 or beyond.
The immediate priority is determining whether the decay exponent α ≈ 1.678 is universal across different gate sets, branching factors, and initial conditions, or whether it is an artifact of our specific circuit. If universal, it may be a fundamental constant of quantum genealogy.