Sharp Complete Modified Log-Sobolev Inequalities on Classical and Quantum Tori

Abstract: We prove that the heat semigroup on the circle has optimal complete modified logarithmic Sobolev constant (1). The proof is based on a matrix-valued Wirtinger inequality and yields the stronger Bogoliubov--Kubo--Mori Fisher information contraction with rate (e^{-2t}). As a consequence, the complete tensorization and transference principle yields the same sharp constant for the heat semigroups on classical and quantum tori.