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Body: Systems with timedependent Hamiltonians are not constrained to energy conservation. Hence, the dynamics of driven physical systems can be significantly different to its static counterpart. Then, driven systems are a new platform where exciting phenomena can be discovered. However, analytical solutions of the dynamics of driven systems are extremely rare and it is typically difficult to gain intuition. Quantum Floquet theory provides a tractable mathematical framework to study periodically driven systems in the form of perturbative expansions in the driving frequency, which is the cornerstone of Floquet engineering. 
Body: The field of quantum information takes a pragmatic approach to examining and utilizing quantum mechanics, seeking to obtain rigorous understandings of which information processing tasks can (e.g. quantum computing, communication) or cannot (e.g. nocloning theorem) be accomplished according to the laws of Nature. At the heart of these desired tasks is the manipulation of various useful quantum features, most prominent examples being entanglement and coherence, which emerge as valuable “resources” that are needed to empower advantages over classical methods. In practice, a particularly important and widelystudied kind of manipulation is to “purify” the quantum resources, since they are inevitably contaminated by noises and thus often lost their power or become unreliable for direct usage. We prove fundamental limitations on how effectively generic noisy resources can be purified enforced by quantum mechanics, which universally apply to any reasonable kind of quantum resource. Remarkably, it is impossible to achieve perfect resource purification, even probabilistically. Our theorems indicate strong limits on the efficiency of distillation, a widelyused type of resource purification subroutine that underpins many key applications of quantum information science. In particular, we give explicit lower bounds on the cost of magic state distillation, a leading scheme for realizing scalable faulttolerant quantum computing. This is the first rigorous understanding of the cost required for practical quantum computing to our knowledge. 




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Body: The average dissipation generated during a slow thermodynamic process can be characterised by introducing a metric on the space of Gibbs states, in such a way that minimallydissipating protocols correspond to geodesic trajectories. Furthermore, the dissipation is proportional to the work fluctuations for classical systems (which follows from the fluctuationdissipation relation (FDR)), so that minimising dissipation also minimises fluctuations. In this talk, I will explain how this geometric picture is modified in the quantum regime. First, I will show that slowly driven quantum systems violate the classical FDR whenever quantum coherence is generated along the protocol, implying that quantum noncommutativity prohibits finding slow protocols that minimise both dissipation and fluctuations simultaneously. Instead, we develop a quantum geometric framework to find processes with an optimal tradeoff between the two quantities. Furthermore, I will show that such quantum fluctuations lead to a nonGaussian work distribution, in contrast to the Gaussian shape typically found in classical slow processes. This talk is based on: arXiv:1810.05583, arXiv:1905.07328, and arXiv:1911.04306. 





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