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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 R\\'enyi 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. 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