Sharp quantum logarithmic Sobolev inequalities via semigroup interpolation

We develop a semigroup interpolation method for sharp logarithmic Sobolev inequalities using spectral calculus and estimates of $\mathrm{R}\mathrm{e}⟨Lx,x^2{⟩}_2$, where $L$ generates an auxiliary symmetric quantum Markov semigroup. We apply the method to free groups and to crossed products of a finite von Neumann algebra $\mathcal{N}$ by $\mathbb{Z}$ or ${\mathbb{Z}}_n$, for $n=2$ or $n\ge 4$, when the action commutes with the semigroup on $\mathcal{N}$. In particular, the word-length generators on quantum tori and the Weyl word-length generators on $M_n(\mathbb{C})$ satisfy logarithmic Sobolev inequalities with optimal constant $2$.