Bimodule Quantum Markov Semigroups

Abstract: We present a systematic investigation of bimodule quantum Markov semigroups within the framework of quantum Fourier analysis. Building on the structure of quantum symmetries, we introduce the concepts of bimodule equilibrium and bimodule detailed balance conditions, which not only generalize the classical notions of equilibrium and detailed balance but also expose interesting structures of quantum channels. We demonstrate that the evolution of densities governed by the bimodule quantum Markov semigroup is the bimodule gradient flow for the relative entropy with respect to quantum symmetries. Consequently, we obtain bimodule logarithmic Sobelov inequalities and bimodule Talagrand inequality with respect to a hidden density from higher dimensional structure. Furthermore, we establish a bimodule Poincar\'{e} inequality for irreducible inclusions and relative ergodic bimodule quantum semigroups.