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Open quantum systems, measurement and control

We live today the second quantum revolution, where the quantum properties such as coherent superposition and entanglement are used in a controlled and reproducible manner to develop new technological tools in computation, communication and high-precision measurement. After a brief discussion on particular features of quantum systems with respect to classical ones (composite systems modeled by tensor products, and the irreversible and unpredictable nature of quantum measurements), I will overview the main mathematical models behind these systems and some of their properties. These models include the discrete Markov chain models for quantum systems under measurement, the continuous-time stochastic master equations, and the Lindblad-type master equation modelling the average evolution. I will also briefly discuss some control and stabilization problems which are at the center of the current developments for instance in the area of quantum error correction.

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