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

Abstract

In the study of Markovian processes, one of the principal achievements is the equivalence between the $\Phi$-Sobolev inequalities and an exponential decrease of the $\Phi$-entropies. In this work, we develop a framework of Markov semigroups on matrix-valued functions and generalize the above equivalence to the exponential decay of matrix $\Phi$-entropies. This result also specializes to spectral gap inequalities and modified logarithmic Sobolev inequalities in the random matrix setting. To establish the main result, we define a non-commutative generalization of the carré du champ operator, and prove a de Bruijn's identity for matrix-valued functions. The proposed Markov semigroups acting on matrix-valued functions have immediate applications in the characterization of the dynamical evolution of quantum 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 time goes on. Consequently, we obtain a stronger notion of monotonicity of the Holevo quantity - the Holevo quantity of the quantum ensemble decays exponentially in time and the convergence rate is determined by the modified log-Sobolev inequalities. However, in the latter, the matrix $\Phi$-entropy of the quantum ensemble that undergoes the phase-damping Markovian evolution generally will not decay exponentially. This is because there are multiple equilibrium states for such a channel. Finally, we also consider examples of statistical mixing of Markov semigroups on matrix-valued functions. We can explicitly calculate the convergence rate of a Markovian jump process defined on Boolean hypercubes, and provide upper bounds of the mixing time on these types of examples.

References in corpus (24)

• Quantum Metropolis Sampling
• Open Quantum Random Walks
• Monotonicity of quantum relative entropy revisited
• Jensen's Operator Inequality
• Contractivity of positive and trace preserving maps under $L_p$ norms
• Quantum logarithmic Sobolev inequalities and rapid mixing
• The ${\chi }^2$-divergence and Mixing times of quantum Markov processes
• Unital Quantum Channels - Convex Structure and Revivals of Birkhoff's Theorem
• Entropies, convexity, and functional inequalities
• A Hypercontractive Inequality for Matrix-Valued Functions with Applications to Quantum Computing and LDCs
• Open Quantum Walks on Graphs
• Hypercontractivity of quasi-free quantum semigroups
• Binomial-Poisson entropic inequalities and the M/M/$\infty$ queue
• Spectral convergence bounds for classical and quantum Markov processes
• A Cutoff Phenomenon for Quantum Markov Chains
• Perturbation Bounds for Quantum Markov Processes and their Fixed Points
• Entropy Production of Doubly Stochastic Quantum Channels
• Subadditivity of Matrix phi-Entropy and Concentration of Random Matrices
• Spectral Transition for Random Quantum Walks on Trees
• Quantum reverse hypercontractivity
• Hypercontractivity and the logarithmic Sobolev inequality for the completely bounded norm
• Characterisations of Matrix and Operator-Valued $\Phi$-Entropies, and Operator Efron-Stein Inequalities
• Generators of Quantum Markov Semigroups
• The Learnability of Unknown Quantum Measurements

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