Lean-QIT: Towards a Formal Infrastructure for Quantum Information Theory

2026-07-10

AI Summary

This work addresses the absence of a unified, reusable operational framework in quantum information theory for formally capturing encodings, error criteria, and capacity notions— a gap that has impeded machine verification of foundational theorems. To bridge this, the authors develop LeanQIT, a formal library in Lean 4 that, for the first time, decouples operational semantics from information-theoretic characterizations, offering composable and kernel-verified interfaces. The library enables modular formalization of quantum states, channels, encodings, hypothesis testing, and both single-shot and asymptotic analyses. It has been successfully employed to verify key results including Schumacher’s source coding theorem, the Holevo–Schumacher–Westmoreland classical capacity theorem, its entanglement-assisted variant, and the corresponding strong converse theorem, thereby laying a rigorous foundation for AI-assisted reasoning in quantum information theory.

Abstract

Quantum information theory (QIT) characterizes the capabilities and fundamental limits of quantum information processing, underpinning quantum communication, computation, and error correction. Formalizing its coding theorems requires connecting finite-block protocols, analytic inequalities, and asymptotic limits within a unified machine-checked framework. Existing developments, however, lack a reusable operational layer that defines codes, error criteria, achievable rates, and capacities independently of their information-theoretic characterizations. In this work, we present LeanQIT, a Lean 4 library for finite-dimensional QIT. It provides composable, kernel-checked interfaces for quantum states and channels, source and channel codes, finite-block performance criteria, hypothesis testing, one-shot quantities, and asymptotic rate constructions. Using this infrastructure, we formalize Schumacher's quantum source-coding theorem, the Holevo--Schumacher--Westmoreland classical-capacity theorem, and the entanglement-assisted classical-capacity theorem together with its strong converse. By separating operational definitions from analytic characterizations and exposing reusable achievability, converse, and asymptotic components, Lean-QIT provides a machine-readable foundation for formal QIT and a compositional knowledge substrate for emerging AI-assisted formalization, automated proof search, and agentic reasoning in quantum information and computation.

Problem

• quantum information theory
• formal verification
• operational layer
• machine-checked framework
• coding theorems

Innovation

• formal verification
• quantum information theory
• Lean 4
• composable infrastructure
• machine-checked proofs