# ⚛  L1 Principle — Entangled-Photon Microscopy (N00N / SPDC imaging)

**ID:** `L1-155` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Quantum Imaging — *Phase sensitivity beyond shot-noise via N-photon entanglement*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** 2D phase
> **📡 Carrier:** photon · **🌫 Noise:** shot poisson
> **⚖ Difficulty (δ):** 10 · **⛓ Block:** 41554241

---

## 🧠 1. Introduction

**Entangled-Photon Microscopy (N00N / SPDC imaging)** is a **nonlinear inverse problem** whose unknown lives in **2D phase** space, within the **Phase sensitivity beyond shot-noise via N-photon entanglement** sub-domain of **Quantum Imaging**.

Measurements consist of photons collected by an optical detector via a **entangled photon imaging** sensing mechanism.

The forward operator applies, in order: L · spdc source operator; L · n photon interference operator; D · coincidence count operator; integration over the solid angle of incidence/emission.

Observations are corrupted by Poisson shot noise from quantum-limited detection. Existence of the recovered 2D phase is guaranteed within the declared Omega bounds. Uniqueness is local rather than global (non-convex landscape); convergence depends on initialisation and priors. Stability is moderately conditioned (kappa_eff ~= 30); photon_loss dominates the stability cliff; mode_coupling_efficiency and the remaining mismatch parameters contribute higher-order bias terms. Photon-shot-noise-limited (poisson counting) sets the irreducible data-fidelity floor, while TV / wavelet-sparsity / deep priors stabilise recovery at the ill-conditioned end of Omega.

## ⚙ 2. Forward Model

Physical chain: **x** → L · spdc source → L · n photon interference → D · coincidence count → Angular integration → **y** (detector).

```
y = ∫dΩ `D.coincidence_count` `L.n_photon_interference` `L.spdc_source` x,    measurements ~ Poisson(αy)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.spdc_source` | L · spdc source operator |
| `L.n_photon_interference` | L · n photon interference operator |
| `D.coincidence_count` | D · coincidence count operator |
| `int.angular` | Integration over the solid angle of incidence/emission |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Quantum Imaging |
| Sub domain | Phase sensitivity beyond shot-noise via N-photon entanglement |
| Carrier | photon |
| Problem class | nonlinear_inverse |
| Solution space | 2D_phase |
| Noise model | shot_poisson |
| Integration axis | spatial |
| Difficulty delta | 10 |
| L dag | 4.5 |

## 📡 4. Measurement Model

Existence of the recovered 2D phase is guaranteed within the declared Omega bounds. Uniqueness is local rather than global (non-convex landscape); convergence depends on initialisation and priors. Stability is moderately conditioned (kappa_eff ~= 30); photon_loss dominates the stability cliff; mode_coupling_efficiency and the remaining mismatch parameters contribute higher-order bias terms. Photon-shot-noise-limited (poisson counting) sets the irreducible data-fidelity floor, while TV / wavelet-sparsity / deep priors stabilise recovery at the ill-conditioned end of Omega.

| Metric | Value |
|---|---|
| Metric | PSNR_dB |
| Secondary | SSIM |

## 📏 5. Operating Range (Ω)

**Center problem class:** `entangled_photon_microscopy` · **Forward operator:** `entangled_photon_microscopy_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 256 |
| W | px | 256 |
| N photons | photons | 2 |
| Photon loss | — | 0.3 |
| N coincidences | — | 100000 |
| Phase instability | — | 0 |
| Detector dark counts | — | 0.01 |
| Mode coupling efficiency | — | 0.5 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 64 |
| W | px | 64 |
| N photons | photons | 2 – 10 |
| Photon loss | — | 0.0 – 0.9 |
| N coincidences | — | 1000 – 100000000 |
| Phase instability | — | 0.0 – 0.5 |
| Detector dark counts | — | 0.0 – 1.0 |
| Mode coupling efficiency | — | 0.1 – 1.0 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 16.0

| Metric | Range |
|---|---|
| Psnr db | 5.0 – 40.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `600` and δ = `10`.

## 💾 8. Reference Dataset

- **primary** · weight 1.0 · IPFS _(not pinned yet)_

## 9. On-chain Registration

- **Chain hash:** `0x2462745509ce83bcf98cd62c882c8843238dd7818bf8c2aef6892b03fdd06d09`
- **Chain tx hash:** `0x6a186a8078f676549c0981d4dd7c97fbd2449fcdc2187a16981c6389b131f0d5`
- **Chain block:** `41554241`

---

## File Mapping

This bundle consists of: `L1-155.md`, `L1-155.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-155.md` | Source of truth — edit this | Human or LLM |
| `L1-155.json` | Structured metadata for the registry | LLM regenerates from the sections above |

**Prompt for your LLM after editing this Markdown:**

> Read the attached Markdown. Regenerate the sibling `.json` so every field matches.
> Preserve the schema documented in the rows above.
> Output each file in its own fenced code block tagged with the filename.
> Output only the JSON object.

_This Markdown was auto-synthesized from the catalog row for `L1-155`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._