Logarithmic Sobolev inequalities (LSI) first were introduced by Gross in 1970s as an equivalent formulation of hypercontractivity. LSI have been well studied in the past few decades and found applications to information theory, optimal transport, and graphs theory. Recently matrix-valued LSI have been an active area of research. Matrix-valued LSI of Lindblad operators are closely related to decoherence of open quantum systems. In this talk, I will present recent results on matrix-valued LSI, in particular a geometric approach to matrix-valued LSI of Lindblad operators. This talk is based on joint work with Li Gao, Marius Junge, and Nicholas LaRacuente.