Logarithmic Sobolev inequalities have been well studied in the last few decades. We consider the matrix-valued logarithmic Sobolev inequalities and recapture the celebrated Bakry-\'Emery criterion. Using tools from combinatorics, we obtain computable lower bounds for matrix-valued log-Sobolev inequalities of Lindblad operators associated to graph H\"ormander systems. Inspired by quantum information theory, we can show that those estimates apply to a larger family of matrix-valued Sobolev type inequalities. This is joint work with Li Gao, Marius Junge, and Nicholas LaRacuente.
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