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Spectral methods for quantum Markov chains

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Quantum Markov chains constitute a natural quantum-mechanical generalization of the concept of a Markov chain and are used to model the evolution of certain quantum systems. Both in the classical and quantum context many applications require estimates that quantify convergence time and stability of the chain. We develop a powerful framework that yields such estimates in terms of the spectrum of the transition map. The methods employed are new even to the well studied theory of classical Markov chains. The key observation is to relate the original problem of bounding a function of a transition map of a Markov chain to a Nevanlinna-Pick interpolation problem in Wiener algebra. We provide an introduction to an operator theoretic approach to the Nevanlinna-Pick problem employing Ando’s Theorem and other strong results from the theory of Hilbert function spaces.

This talk is part of the CQIF Seminar series.

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