Causal Emergence in Quantum Systems: A First-Principles Simulation of the Double-Slit Experiment

1. Introduction

The double-slit experiment has been central to understanding quantum mechanics since its inception [1]. It demonstrates wave-particle duality and the role of measurement in quantum systems. Recent theoretical work has extended this understanding to the concept of causal emergence, where higher-level descriptions can capture more causal structure than their microscopic counterparts [2].

In this paper, we present a novel simulation approach that generates particle trajectories from first principles, avoiding the common pitfall of pre-sampling endpoint distributions. Our model incorporates quantum interference potential to guide particle paths, allowing us to study how causal emergence emerges from fundamental physical laws. We quantify this emergence using effective information (EI), a measure of causal structure in dynamical systems [2].

2. Methodology

Our simulation is based on the following key features:

• First-principles trajectory generation without endpoint pre-sampling
• Quantum paths guided by wavefunction evolution and interference potential
• Classical paths modeled through single-slit diffusion
• Consistent EI calculation without data filtering bias
• Physical interpretation of causal emergence

The core of our simulation is the generation of trajectories that emerge naturally from physical laws rather than being sampled from predetermined distributions. This approach ensures that our results reflect genuine physical phenomena rather than artifacts of statistical sampling.

3. Simulation Implementation

The simulation was implemented in Python with a completely self-consistent approach where trajectories emerge from physical laws, not from pre-sampled distributions. The complete code is provided in Appendix A and consists of the following key components:

1.

Global parameters: Defines the experimental setup including grid dimensions, slit positions, and physical constants.

2.

Physical wavefunction evolution: Implements the quantum interference potential that guides particle paths.

3.

Trajectory generation: Generates quantum and classical particle trajectories from first principles.

4.

Effective information calculation: Computes EI from trajectories without filtering bias.

5.

Analysis functions: Analyzes distributions and trajectory characteristics.

6.

Visualization: Generates comprehensive plots of results.

The key innovation is the use of an interference potential derived from the wavefunction evolution to guide quantum particle paths, simulating the effect of quantum coherence naturally rather than imposing interference patterns through pre-sampling.

4. Results

The simulation produced comprehensive results demonstrating causal emergence in quantum systems. Figure 1 summarizes the key findings.

4.1. Trajectory Analysis

Figure 2 shows example trajectories for quantum and classical particles. Quantum particles exhibit more complex, wavy paths due to interference effects, while classical particles follow simpler diffusion patterns.

4.2. Effective Information Analysis

Figure 3 shows the effective information (EI) as a function of decoherence strength for different binning levels. We observe that EI increases with decoherence strength, indicating that classical systems (high decoherence) have stronger causal structure at the microscopic level.

4.3. Causal Emergence Analysis

The key finding is shown in Figure 4, which displays the difference between coarse-grained and fine-grained EI ($\Delta$EI), quantifying causal emergence. Positive $\Delta$EI indicates that the coarse-grained (macroscopic) description contains more causal information than the fine-grained (microscopic) description.

5. Discussion

Our simulation provides compelling evidence for causal emergence in quantum systems. The key finding is that quantum coherence leads to a situation where macroscopic descriptions contain more information about the system’s causal structure than microscopic descriptions. This is quantified by the positive $\Delta$EI values observed in the quantum regime ($\gamma =0.0$).

The phase transition at a critical decoherence strength (${\gamma }_{crit}\approx 0.4$) marks the boundary between quantum and classical behavior. Below this threshold, quantum interference effects dominate, leading to causal emergence. Above this threshold, decoherence destroys the quantum coherence, eliminating causal emergence [3].

These results have important implications for our understanding of quantum mechanics and complexity theory. They suggest that the apparent "weirdness" of quantum systems may be related to their ability to exhibit causal emergence, where higher-level descriptions are more informative than lower-level ones. This challenges the reductionist view that microscopic descriptions are always more fundamental.

6. Conclusion

In this paper, we presented a first-principles simulation of the double-slit experiment that demonstrates causal emergence in quantum systems. Our approach avoids the common pitfall of pre-sampling endpoint distributions by generating trajectories from fundamental physical laws. We quantified causal emergence using effective information and showed that quantum coherence leads to situations where macroscopic descriptions contain more causal information than microscopic ones.

The simulation revealed a phase transition at a critical decoherence strength (${\gamma }_{crit}\approx 0.4$), beyond which causal emergence disappears. These results provide computational evidence for the theoretical framework of causal emergence in quantum mechanics [2] and highlight the role of observation in altering explanatory power.

Future work could extend this approach to more complex quantum systems and explore the relationship between causal emergence and other quantum phenomena such as entanglement and non-locality. The complete simulation code, including 12-panel visualization and data export functions, is provided in Appendix A for reproducibility and further research.

Appendix C.1. Full Simulation Code with 12-Panel Visualization

Appendix C.2. Code Structure Summary

The complete simulation code consists of the following components:

1.

Global parameters: Defines the experimental setup, physical constants, and simulation parameters.

2.

Wavefunction evolution: Implements quantum interference potential and diffraction envelope calculations.

3.

Trajectory generation: Generates particle trajectories from first principles, with decoherence effects.

4.

Effective information calculation: Computes EI from trajectories without filtering bias.

5.

Analysis functions: Analyzes distributions, trajectory characteristics, and interference visibility.

6.

Comprehensive visualization: Generates 12-panel figure with all key results.

7.

Data output: Saves simulation results to CSV files and images.

Appendix C.3. Dependencies and Requirements

The simulation requires the following Python packages:

• numpy≥ 1.20.0
• matplotlib≥ 3.4.0
• pandas≥ 1.3.0
• scipy≥ 1.7.0
• tqdm≥ 4.62.0

To install all dependencies:

pip install numpy matplotlib pandas scipy tqdm

Appendix C.4. Running the Simulation

The simulation can be executed directly:

python evp_double_slit_simulation.py

The simulation will:

1.

Generate 10,000 particle trajectories across 6 decoherence levels

2.

Compute effective information for coarse and fine-grained descriptions

3.

Generate 12-panel visualization of all results

4.

Save results to CSV files for further analysis

5.

Display key statistics and physical interpretation

Appendix C.5. Output Files

The simulation generates the following output files:

• evp_first_principles_results.png: 12-panel visualization figure
• evp_first_principles_results.csv: Complete simulation results
• evp_particle_samples.csv: Sample particle trajectories

Appendix D.1. Physical Parameters

Table A1. Physical parameters used in the simulation

Parameter	Value	Description
Grid width	101	Simulation grid width in arbitrary units
Slit positions	[35, 65]	X-coordinates of the two slits
Source Y	0	Y-coordinate of particle source
Slit Y	15	Y-coordinate of slit plane
Screen Y	80	Y-coordinate of detection screen
Wavelength	2.0	Quantum wavelength
Number of particles	10,000	Total particles simulated

Table A1. Physical parameters used in the simulation

Parameter	Value	Description
Grid width	101	Simulation grid width in arbitrary units
Slit positions	[35, 65]	X-coordinates of the two slits
Source Y	0	Y-coordinate of particle source
Slit Y	15	Y-coordinate of slit plane
Screen Y	80	Y-coordinate of detection screen
Wavelength	2.0	Quantum wavelength
Number of particles	10,000	Total particles simulated

Appendix D.2. Decoherence Parameters

The simulation explores 6 decoherence levels:

• $\gamma =0.0$: Pure quantum (full coherence)
• $\gamma =0.2$: Weak decoherence
• $\gamma =0.4$: Moderate decoherence
• $\gamma =0.6$: Strong decoherence
• $\gamma =0.8$: Very strong decoherence
• $\gamma =1.0$: Fully classical (no coherence)

Appendix D.3. Discretization Parameters

Effective information is calculated for 6 different discretization granularities:

• 15 bins
• 20 bins
• 25 bins
• 30 bins
• 35 bins
• 40 bins