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

Kazumi Kasaura, Kei Tsukamoto, Kento Mori, Risa Mizuno, Takahiro Namatame, Yuta Oriike, Masaya Taniguchi, Sho Sonoda, Hayata Yamasaki

Featured July 18, 2026

Original Paper

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Simply

A new digital library for quantum math proofs helps computers check if complex quantum theories are absolutely correct, making it easier for future AI to assist in scientific discoveries.

In depth
The paper introduces a Lean 4 library that provides a machine-checkable foundation for quantum information theory. It achieves this by developing a basis-independent operator-theoretic framework for finite-dimensional quantum systems and formalizing a comprehensive hierarchy of noncommutative trace inequalities. As a central demonstration, the authors present a novel, formalized proof of the Data Processing Inequality (DPI) for the sandwiched Rényi relative entropy, using an alternative strategy based on Young inequalities instead of traditional Euler-Lagrange optimization.

Key Takeaways

  • 1
    The authors developed a basis-independent operator-theoretic framework for finite-dimensional quantum systems in Lean 4, enabling coordinate-free statements and reuse of Mathlib's abstract algebra.
  • 2
    A reusable hierarchy of noncommutative trace inequalities was formalized, including Jensen's operator inequality, generalized perspectives, operator power means, and Lieb-Ando trace inequalities.
  • 3
    An alternative proof strategy for the Data Processing Inequality (DPI) of the sandwiched Rényi relative entropy was formalized, leveraging Young and reverse-Young inequalities for variational formulas and separating positive definite from positive semidefinite cases.

Conceptual Flow

HIGH LEVEL
1
Methodology: Building a Verified Quantum Math Library

The authors built a special digital library where every quantum math rule is checked by a computer, making sure it's perfectly correct.

Basic Math Rules
Build on
Quantum System Rules
Operator Rules
Entropy Rules
2
Results: Proving a Key Quantum Rule

Using this library, they proved a big rule about how quantum information changes, showing their computer-checked system works for complex problems.

Quantum Information
Quantum Channel
Process and Compare
Information Never Increases