Decoherence-Dependent Branch Survival in Constrained Many-Worlds: Recovery of the Lindblad Channel from Sequential Measurement Statistics

1. Introduction

The Many-Worlds Interpretation (MWI) of quantum mechanics proposes that the wave function never collapses. Instead, the Schrödinger equation holds universally, and every quantum interaction produces branching of the universal wave function into dynamically independent components corresponding to different macroscopic outcomes.

A persistent challenge for MWI is the probability problem: if all branches are equally real, why do observers experience outcome frequencies consistent with the Born rule? Varshovi (2026) introduced a new approach using measure-theoretic methods from ergodic theory, proving that in long-term consecutive experiments, almost all parallel worlds would contain frequency distributions that statistically refute the Born rule. Only a Lebesgue-null set of branches maintains statistical consistency with Born predictions.

However, Varshovi's framework rests on a critical assumption: the Born probabilities are fixed throughout the measurement sequence. In any real physical setting, quantum systems interact with their environment between measurements. Decoherence modifies the quantum state between successive observations, causing the Born probabilities to drift over time.

This paper addresses a direct question: what happens to branch survival dynamics when the system is open? We answer using Operator Discovery (OD), a method that discovers governing mathematical equations from raw data without assuming a model family a priori.

2. Background

2.1 Constrained Many-Worlds and Born Rule Statistics

In standard MWI, a measurement with n possible outcomes on a system in superposition produces n branches. In a sequence of N measurements, the total branches grow as nⁿ. Varshovi defines the subset B as those branches in which observed frequencies converge to Born rule probabilities. Using Birkhoff's ergodic theorem, he proves the complement of B has full Lebesgue measure. The set B is a Lebesgue-null set, though still uncountable.

2.2 The Missing Piece: Decoherence

In open quantum systems, between measurements the system evolves under the Lindblad master equation:

This evolution changes the density matrix ρ, and consequently the Born probabilities, between successive measurements. The measurement sequence is therefore non-stationary.

2.3 Operator Discovery

Operator Discovery (OD) is a data-driven method for recovering the governing mathematical operators of a dynamical system from observational data. Unlike machine learning approaches that approximate input-output mappings, OD discovers explicit functional forms without assuming a model family.

3. Methods

3.1 Simulated Quantum Systems

Five quantum systems were simulated using QuTiP:

Closed systems: System A (single qubit, asymmetric), System B (single qubit, symmetric), System C (two-qubit Bell state).

Open systems: System D (single qubit, pure dephasing: γ = 0.1), System E (single qubit, amplitude damping: γ = 0.05).

3.2 Measurement Chain Protocol

For each system, long chains of sequential projective measurements were simulated. Closed systems: 1000 chains × 10,000 steps. Open systems: 500 chains × 2000 steps.

4. Results

4.1 Closed Systems: Universal Convergence (Varshovi Confirmed)

All three closed systems exhibit convergence with power-law exponent α ≈ 0.5, consistent with the theoretical prediction from the law of large numbers. The operators are functionally identical across systems, confirming universal closed-system dynamics.

4.2 Open Systems: Three Convergence Regimes

Regime 1 — Convergent (closed): α ≈ 0.5. Deviation decreases as 1/√n.

Regime 2 — Locked (pure dephasing): α = 0.0. Deviation is perfectly flat. The convergence operator is the identity.

Regime 3 — Divergent (amplitude damping): α = −0.114. Deviation increases over time due to deterministic population drift toward the ground state.

4.3 Recovery of the Lindblad Channel

OD recovers the decoherence operators without knowledge of the Lindblad operators. For amplitude damping, the fixed point p* = 1.0 correctly identifies the ground state relaxation target.

4.4 Gamma Sweep: Quantitative Recovery

A sweep over the amplitude damping rate γ from 0.02 to 1.0 validates quantitative recovery:

A meta-operator fit yields G_slope(γ) = exp(−0.452γ), compared to the theoretical exp(−0.500γ). Fit quality: R² = 0.997.

4.5 Transfer Tests

Closed → Closed: R² = 0.947. High transfer confirms universal closed-system dynamics.

Open → Open: R² = 0.701. Moderate transfer between structurally different open systems.

Closed → Open: Catastrophic failure. R² effectively zero.

5. Discussion

5.1 Branch Survival is Physics, Not Just Statistics

In closed systems, pruning is purely statistical, driven by the law of large numbers. In open systems, pruning acquires physical content: the Lindblad decoherence channel determines the convergence regime, the rate of pruning, and the attractor state. The decoherence operator is recoverable from the measurement record alone.

5.2 Connection to Quantum Darwinism

These results complement Zurek's quantum Darwinism program. Both questions receive the same answer: the decoherence channel decides. The Lindblad operator serves as the selection mechanism in both frameworks.

5.3 Practical Implications

OD-based decoherence characterization from sequential single-basis measurements offers a simpler alternative to state or process tomography. The gamma sweep demonstrates recovery of coupling strength with R² = 0.997 across two orders of magnitude.

5.4 Limitations

Systems studied are single- and two-qubit with Markovian decoherence. Scaling to larger systems is unvalidated. The 9.7% systematic bias in the recovered exponential coefficient indicates finite-sample effects. Results are interpretation-neutral; MWI framing follows Varshovi's original framework.

6. Conclusion

Branch survival dynamics in constrained Many-Worlds are governed by the physical decoherence environment, extending Varshovi's measure-theoretic result to open quantum systems. Operator Discovery identifies three convergence regimes determined by the Lindblad structure and recovers the exact amplitude damping channel from measurement statistics alone. The pruning operator is the Lindblad channel.

References