optimal-stellar-rank-photon-catalysis

name	optimal-stellar-rank-photon-catalysis
description	Methodology for provably optimal generation of non-Gaussian quantum states (squeezed cat states) via photon catalysis, characterized using stellar rank formalism — enables systematic comparison of fidelity against theoretical maximum for given non-Gaussian resources.

Optimal Stellar Rank Approximation of Squeezed Cat States with Photon Catalysis

Description

A methodology for analyzing and optimizing photon catalysis protocols for generating non-Gaussian quantum states (specifically squeezed coherent state superpositions / "cat states") in bosonic platforms. Uses the stellar rank formalism to characterize the non-Gaussian complexity of both input resources and generated states, enabling provable optimality claims. Identifies parameter regimes for high-fidelity approximations with minimal resources, and benchmarks against Gaussian boson sampling-inspired protocols. Based on arXiv:2607.02427 (Nauth, Walk & Datta, 2026).

Activation Keywords

• stellar rank photon catalysis
• squeezed cat state generation
• non-Gaussian state optimization
• photon catalysis protocol
• bosonic quantum state generation
• stellar rank formalism
• 光子催化生成猫态
• 恒星秩形式化
• non-Gaussian resource complexity

Tools Used

• exec: Run Python scripts for stellar rank computation, fidelity optimization, and loss modeling
• read: Read quantum state definitions, stellar rank specifications
• write: Output optimized protocol parameters and fidelity analysis

Usage Patterns

Cat State Generation Design

Given target squeezed cat state parameters, find optimal photon catalysis protocol achieving maximum fidelity for given non-Gaussian resource budget.

Resource Complexity Analysis

Compare different non-Gaussian state generation protocols by their stellar rank — determine if a protocol is provably optimal or has room for improvement.

Loss-Tolerant Protocol Design

Model experimental imperfections (losses across all optical modes) and identify parameter regimes robust to realistic conditions.

Instructions for Agents

Step 1: Characterize Input Resources via Stellar Rank

1. Define stellar rank: The stellar rank of a quantum state is the number of zeros of its Husimi Q-function in phase space. For Fock state |n⟩, stellar rank = n. For squeezed vacuum, stellar rank = 0 (Gaussian).
2. Compute combined stellar rank: For photon catalysis between |n⟩ (Fock) and |sqz⟩ (squeezed), the combined input stellar rank = n (only the Fock state contributes non-Gaussianity).
3. Characterize measurements: Post-selection measurements also carry stellar rank — include in total resource accounting.

Step 2: Construct Photon Catalysis Protocol

1. Setup: Interfere a low-number Fock state |n⟩ with a squeezed state |ξ⟩ on a beamsplitter.
2. Post-select: Measure one output port in the Fock basis, conditioning on a specific outcome.
3. Output state: The unmeasured port yields a non-Gaussian state — typically a squeezed coherent state superposition (cat state).

Step 3: Evaluate Fidelity Against Stellar Rank Bound

1. Compute maximum achievable fidelity: For any protocol with the same total stellar rank of inputs, compute the theoretical maximum fidelity to the target state.
2. Compare catalysis fidelity: Calculate the actual fidelity achieved by the photon catalysis protocol.
3. Optimality proof: If catalysis fidelity equals the stellar rank bound, the protocol is provably optimal — no other protocol with the same non-Gaussian resources can achieve higher fidelity.

Step 4: Benchmark Against Alternatives

Compare photon catalysis with:

• Gaussian boson sampling-inspired protocols
• Direct heralded generation protocols
• Deterministic Fock state sources

Metrics: success probability, state quality (fidelity), resource complexity (stellar rank).

Step 5: Model Experimental Imperfections

1. Loss modeling: Apply beam splitter model for loss in each optical mode using Fock basis truncation.
2. Robustness analysis: Vary loss parameters and compute fidelity degradation curves.
3. Identify robust regimes: Find parameter combinations where fidelity remains high (>0.95) under realistic loss rates (1-5%).

Key Mathematical Concepts

Stellar Rank Formalism

The stellar function of a pure state |ψ⟩ is: $$S_ψ(z) = ⟨z^*|ψ⟩$$ where |z⟩ is a coherent state. The stellar rank r is the number of zeros of S_ψ(z) in the complex plane.

• Gaussian states: r = 0
• Single photon: r = 1
• Fock state |n⟩: r = n
• Cat state |α⟩ + |-α⟩: r = 2 (two zeros in phase space)

The stellar rank is non-increasing under Gaussian operations and additive under tensor products.

Photon Catalysis Transformation

The beamsplitter transformation between |n⟩ and |ξ⟩ followed by Fock measurement |m⟩: $$|ψ_{out}⟩ ∝ ⟨m|_{BS} BS(θ)(|n⟩ ⊗ |ξ⟩)$$

The output stellar rank ≤ n + m, providing a hard bound on achievable non-Gaussianity.

Optimality Criterion

A protocol is provably optimal if: $$F_{catalysis}(target) = max_{all protocols with same stellar rank} F(protocol, target)$$

This requires computing the maximum fidelity achievable by any state with stellar rank ≤ r to the target — a convex optimization over the stellar rank submanifold.

Error Handling

No Optimal Regime Found

If catalysis doesn't achieve stellar rank bound:

• Increase input Fock number n
• Optimize beamsplitter angle θ
• Consider multi-stage catalysis (sequential catalysis steps)

Loss Too High

If fidelity degrades significantly under realistic losses:

• Use higher squeezing to compensate
• Switch to deterministic Fock state sources (better loss tolerance)
• Apply error mitigation via post-selection filtering

Target State Too Complex

If target stellar rank exceeds input resources:

• Cannot achieve high fidelity — stellar rank is a hard limit
• Either increase input resources or accept lower-fidelity approximation
• Consider approximate targets with lower stellar rank

Examples

Example 1: Squeezed Cat State |α⟩ + |-α⟩

Input:

• Fock state |1⟩ (stellar rank 1)
• Squeezed state |ξ⟩ with r = 0.5 (stellar rank 0)
• Beamsplitter angle θ = π/4

Output:

• Achieved fidelity to target cat state: 0.97
• Stellar rank bound: 0.97 (protocol is provably optimal)
• Success probability: 0.15

Example 2: Loss Analysis

Input:

• Same protocol as Example 1
• 3% loss per mode

Output:

• Fidelity after loss: 0.94
• Still above 0.9 threshold for fault-tolerant applications
• Optimal regime identified for near-term photonic implementations

Resources

• arXiv:2607.02427 — "Optimal stellar rank approximation of squeezed cat states with photon catalysis" (Nauth, Walk & Datta, 2026)
• Stellar rank theory for quantum optics (Bartlett & Sanders framework)
• Gaussian boson sampling and non-Gaussian state generation

Related Skills

• quantum-error-correction-methods — bosonic QEC using cat states
• generalized-kerr-cat-qubit-codes — cat-state-based quantum error correction
• optimal-stellar-rank-photon-catalysis — umbrella skill
• quantum-optical-neuron — optical quantum state processing