Algorithmic Bohmian Mechanics

---
name: algorithmic-bohmian-mechanics
description: "Algorithmic Bohmian Mechanics (aBM) methodology using algorithmic randomness to formulate the distribution postulate as an objective constraining law. Guarantees standard Born statistics for canonical quantum experiments in the limit. Use for quantum foundations, interpretation of quantum mechanics, and algorithmic randomness in physical theories."
metadata:
  arxiv_id: "2606.16165"
  published: "2026-06-15"
  authors: "Jeffrey A. Barrett, Eddy Keming Chen, Josiah Lopez-Wild"
  tags: [quantum-foundations, bohmian-mechanics, algorithmic-randomness, interpretation]
---

# Algorithmic Bohmian Mechanics (aBM)

## Description

Uses algorithmic randomness theory to reformulate Bohmian mechanics' distribution postulate as an objective constraining law, guaranteeing standard Born statistics for canonical quantum experiments in the limit rather than merely with high probability.

## Activation Keywords
- bohmian mechanics, pilot wave theory
- distribution postulate, algorithmic randomness
- quantum foundations, born rule derivation
- typicality condition, quantum probability
- algorithmic bohmian, kolmogorov complexity quantum
- 量子力学基础, 玻姆力学

## Core Framework

### The Problem

Bohmian mechanics requires a special statistical boundary condition (the distribution postulate) to recover the Born rule |ψ|². Standard approaches justify this as a typicality condition — "most" initial configurations yield Born statistics — but this leaves the status of quantum probabilities unclear in a deterministic theory.

### The Solution: Algorithmic Randomness

1. **Algorithmic typicality**: Instead of measure-theoretic typicality, use algorithmic randomness (Martin-Löf randomness) to define admissible initial configurations
2. **Objective constraint**: The distribution postulate becomes a law-like restriction on which initial conditions are physically admissible
3. **Guaranteed convergence**: Born statistics hold in the limit for algorithmically random configurations — not just with high probability, but deterministically for the right class of configurations

### Key Components

| Component | Role |
|-----------|------|
| Algorithmic randomness | Defines admissible quantum states |
| Distribution postulate | Objective constraining law |
| Born statistics | Guaranteed in the limit for admissible states |
| Canonical experiments | Framework applies to standard quantum measurements |

### Mathematical Framework

- Initial state vector treated as Martin-Löf random with respect to |ψ|² measure
- Stochastic relative entropy defined from probability density
- For algorithmically random initial configurations, empirical frequencies converge to Born probabilities
- Sharp typicality condition replaces vague "almost all" statements

## Usage Patterns

### Pattern 1: Quantum Foundations Analysis
When analyzing interpretations of quantum mechanics, use aBM to clarify the status of probabilities in deterministic hidden-variable theories.

### Pattern 2: Algorithmic Randomness Applications
Apply the algorithmic distribution postulate framework to other physical theories requiring statistical boundary conditions.

### Pattern 3: Measurement Problem Resolution
Use aBM's guaranteed Born statistics to address concerns about probability in deterministic quantum theories.

## Error Handling

### Admissibility Criteria
The framework requires specifying admissible quantum-mechanical states and measurements. When applying to novel scenarios, verify that the measurement setup falls within the class of "canonical experiments" covered by the theory.

### Determinism vs Probability
aBM is deterministic at the fundamental level but recovers probabilistic predictions. Do not confuse algorithmic typicality with measure-theoretic probability — the former is strictly stronger.

## References
- arXiv: 2606.16165 — "The Distribution Postulate in Algorithmic Bohmian Mechanics"
- Martin-Löf randomness theory
- Bohmian mechanics foundations