# Quantum Emergence Research Area

*Investigating the intersection of quantum mechanics and computational mechanics*

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## Overview

This research area explores how quantum mechanics can simplify the representation of stochastic processes, building on Mile Gu's seminal work showing that quantum models can be more efficient than classical epsilon-machines.

## Key Questions

1. **Can we re-derive Gu's results?** Implement quantum epsilon-machines in emic and verify the quantum advantage for specific processes.

2. **What is the magnitude of quantum advantage?** For which processes is $C_q \ll C_\mu$? Can we identify structural features that predict large quantum advantage?

3. **Can emic compute crypticity?** The gap $\chi = C_\mu - E$ represents classical waste—can we add this to emic's analysis module?

4. **How do we extend to quantum processes?** What infrastructure is needed to handle quantum states, channels, and measurements?

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## Connection to emic

| emic Concept | Quantum Extension |
|--------------|-------------------|
| `EpsilonMachine` | q-machine (quantum ε-machine) |
| Statistical complexity $C_\mu$ | Quantum complexity $C_q = S(\rho)$ |
| Excess entropy $E$ | Quantum excess entropy $E_Q$ |
| Causal states (classical) | Quantum causal states (non-orthogonal) |
| Crypticity $\chi$ | Reducible via quantum models |

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## Research Phases

### Phase 1: Classical Preparation (Current)
- Implement crypticity computation in emic
- Validate $C_\mu$ and $E$ against literature values
- Identify processes with large crypticity gaps

### Phase 2: Quantum Model Construction
- Implement quantum epsilon-machine representation
- Define quantum causal states for toy models
- Compute $C_q$ and verify $C_q < C_\mu$

### Phase 3: Re-derive Gu's Results
- Reproduce the "perturbed coin" example
- Verify unbounded quantum advantage cases
- Explore thermodynamic connections

### Phase 4: Novel Contributions
- Extend to mixed-state quantum processes
- Study quantum-to-classical emergence
- Connect to quantum thermodynamics

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## Directory Structure

```
quantum-emergence/
├── README.md                    # This file
├── review/                      # Literature review
│   ├── README.md               # Review overview
│   ├── key-papers.md           # Annotated bibliography
│   └── concepts.md             # Key concepts and definitions
├── experiments/                 # Computational experiments
│   └── ...
├── theory/                      # Theoretical development
│   └── ...
└── implementation/              # emic extension design
    └── ...
```

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## Related Specifications

- [Spec 014: Quantum Computational Mechanics](../../specifications/014-quantum-computational-mechanics.md)
- [Roadmap M6-M7](../../plan/ROADMAP.md)

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*Created: 2026-01-27*
