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In the study of Markovian processes, one of the principal achievements is theequivalence between the $\Phi$-Sobolev inequalities and an exponential decreaseof the $\Phi$-entropies. In this work, we develop a framework of Markovsemigroups on matrix-valued functions and generalize the above equivalence tothe exponential decay of matrix $\Phi$-entropies. This result also specializesto spectral gap inequalities and modified logarithmic Sobolev inequalities inthe random matrix setting. To establish the main result, we define anon-commutative generalization of the carr\'e du champ operator, and prove a deBruijn's identity for matrix-valued functions.  The proposed Markov semigroups acting on matrix-valued functions haveimmediate applications in the characterization of the dynamical evolution ofquantum ensembles. We consider two special cases of quantum unital channels,namely, the depolarizing channel and the phase-damping channel. In the former,since there exists a unique equilibrium state, we show that the matrix$\Phi$-entropy of the resulting quantum ensemble decays exponentially as timegoes on. Consequently, we obtain a stronger notion of monotonicity of theHolevo quantity - the Holevo quantity of the quantum ensemble decaysexponentially in time and the convergence rate is determined by the modifiedlog-Sobolev inequalities. However, in the latter, the matrix $\Phi$-entropy ofthe quantum ensemble that undergoes the phase-damping Markovian evolutiongenerally will not decay exponentially. This is because there are multipleequilibrium states for such a channel.  Finally, we also consider examples of statistical mixing of Markov semigroupson matrix-valued functions. We can explicitly calculate the convergence rate ofa Markovian jump process defined on Boolean hypercubes, and provide upperbounds of the mixing time on these types of examples.

Exponential Decay of Matrix $\Phi$-Entropies on Markov Semigroups with Applications to Dynamical Evolutions of Quantum Ensembles