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arXiv · 2510.08672

A Formalization of the Generalized Quantum Stein's Lemma in Lean

Alex Meiburg·Leonardo A. Lessa·Rodolfo R. Soldati
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Abstract

The Generalized Quantum Stein's Lemma is a theorem in quantum hypothesis testing that provides an operational meaning to the relative entropy within the context of quantum resource theories. Its original proof was found to have a gap, which led to a search for a corrected proof. We formalize the proof presented in [Hayashi and Yamasaki (2024)] in the Lean interactive theorem prover. This is the most technically demanding theorem in physics with a computer-verified proof to date, building with a variety of intermediate results from topology, analysis, and operator algebra. In the process, we rectified minor imprecisions in [HY24]'s proof that formalization forces us to confront, and refine a more precise definition of quantum resource theory. Formalizing this theorem has ensured that our Lean-QuantumInfo library, which otherwise has begun to encompass a variety of topics from quantum information, includes a robust foundation suitable for a larger collaborative program of formalizing quantum theory more broadly.

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Alex Meiburg, Leonardo A. Lessa, Rodolfo R. Soldati. 2025-10-09. A Formalization of the Generalized Quantum Stein's Lemma in Lean. https://arxiv.org/abs/2510.08672

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