Lean-Quantum: Toward AI-Assisted Formalization of Quantum Information

2026-07-06

AI Summary

This work addresses the absence of a machine-verifiable formal foundation for key inequalities in quantum information theory, notably the data processing inequality for the sandwiched Rényi relative entropy. Building upon Lean 4 and compatible with Mathlib, we develop a basis-independent formal framework for finite-dimensional quantum mechanics that integrates operator theory, noncommutative trace inequalities, tensor product structures, and the Choi–Kraus–Stinespring representation. Within this framework, we provide the first complete formalization and rigorous machine-checked proof of the data processing inequality for the sandwiched Rényi relative entropy. Our development not only yields a formal derivation of strong subadditivity but also fills a crucial gap in the generalized quantum Stein’s lemma, thereby establishing a verifiable foundation for AI-assisted research in quantum information theory.

Abstract

Quantum information theory is built on entropic quantities; among them, the sandwiched Rényi relative entropy is a fundamental divergence with various applications, and its data processing inequality (DPI) under quantum channels is a cornerstone result. In this work, we present a Lean 4 library for quantum information, designed as a reusable formal infrastructure for theoretical analysis. As a central demonstration of the library, we formalize the DPI for the sandwiched Rényi relative entropy for positive semidefinite operators on finite-dimensional quantum systems. The library provides a basis-independent operator-theoretic framework for finite-dimensional quantum mechanics compatible with the standard mathematical library Mathlib, including reusable interfaces for finite-dimensional systems, states, channels, tensor products, partial traces, Choi operators, Kraus representations, and Stinespring representations. It also builds infrastructure for noncommutative trace inequalities, including operator monotonicity and convexity via the real continuous functional calculus, block-operator positivity, Hilbert-Schmidt operator spaces, Jensen's operator inequality, generalized perspectives, operator power means, and Lieb-Ando trace inequalities. On top of this framework, we formalize entropy-specific ingredients for the DPI: variational formulas for the sandwiched quasi-entropy via Young and reverse-Young inequalities, tensor-product compatibility of real powers, and Haar measures on unitary groups. Together, these components yield a Lean formalization of the DPI, give strong subadditivity as a corollary, and provide the last missing component needed to complete the Lean formalization of the generalized quantum Stein's lemma. More broadly, the development provides machine-checkable foundations for future formalized and AI-assisted research in quantum information theory.

Problem

• quantum information
• data processing inequality
• sandwiched Rényi relative entropy
• formal verification
• Lean 4

Innovation

• Lean 4 formalization
• sandwiched Rényi relative entropy
• data processing inequality
• operator-theoretic framework
• noncommutative trace inequalities