lean-quantum-formal-verification

name	lean-quantum-formal-verification
description	AI-assisted formal verification methodology for quantum information theory using Lean 4 theorem prover
category	quantum
activation	Lean 4, formal verification, quantum information, data processing inequality, sandwiched Renyi entropy, machine-checkable proofs, AI-assisted formalization
arxiv_id	2607.05492

Lean-Quantum Formal Verification

Overview

AI-assisted formal verification methodology for quantum information theory using Lean 4 theorem prover. Formalizes fundamental results in quantum information (DPI for sandwiched Rényi relative entropy) with machine-checkable proofs.

Core Methodology

1. Operator-Theoretic Framework: Basis-independent framework for finite-dimensional quantum mechanics compatible with Mathlib
2. Noncommutative Trace Inequalities: Operator monotonicity, convexity, Jensen's inequality, Lieb-Ando trace inequalities
3. Entropy Formalization: Variational formulas for sandwiched quasi-entropy via Young/reverse-Young inequalities
4. Haar Measure Integration: Haar measures on unitary groups for quantum channel analysis

Key Components

• States & Channels: Choi operators, Kraus representations, Stinespring representations
• Tensor Products: Partial traces, tensor-product compatibility of real powers
• Block Operators: Block-operator positivity, Hilbert-Schmidt operator spaces
• Operator Means: Generalized perspectives, operator power means

Applications

• Formal verification of quantum information protocols
• AI-assisted theorem proving in quantum mechanics
• Machine-checkable quantum error correction proofs
• Formalized quantum Stein's lemma

Verification Steps

1. Formalize quantum states as positive semidefinite operators
2. Define quantum channels via Choi/Kraus/Stinespring representations
3. Prove data processing inequality for sandwiched Rényi entropy
4. Derive strong subadditivity as corollary