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Exponential decay of matrix \(\Phi\)-entropies on Markov semigroups with applications to dynamical evolutions of quantum ensembles. (English) Zbl 1390.81267

The authors propose a construction of Markov semigroups acting on matrix-valued functions instead of single quantum states, with application to dynamical systems built on matrix components. Functional inequalities such as spectral gap and logarithmic Sobolev inequalities are characterized in terms of the exponential decay of matrix \(\Phi\)-entropy functionals of the semigroup, with quantum dynamical semigroups in depolarizing and phase-damping channels considered as examples. In addition, known functional inequalities on the symmetric Boolean hypercube are recovered and extended to matrix-valued functions, and bounds on mixing times are obtained on quantum random graphs.

MSC:

81S22 Open systems, reduced dynamics, master equations, decoherence
60J65 Brownian motion
94A17 Measures of information, entropy
94A40 Channel models (including quantum) in information and communication theory
81Q80 Special quantum systems, such as solvable systems
47A60 Functional calculus for linear operators
81Q35 Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices
05C80 Random graphs (graph-theoretic aspects)
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