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Quantum information theory and quantum computation come from the merging of classical information theory with quantum mechanics. This takes place in a very natural way, by substituting the classical information units (bits) with their quantum analogues (quantum bits, or qubits). Despite the simplicity of such a change, it implies consequences of both fundamental and practical interest. Encoding the information in a quantum object makes it possible to take advantage of the quantum mechanical features of Nature, and this has proved successful in many fields of information and computation theory. In particular, entanglement --i.e. stronger-than- classical correlations between two (or more) quantum systems-- has been proved to be a key resource in many tasks, such as super-dense coding and quantum teleportation. For this reason, the study of entanglement and its properties is a central matter of investigation in the context of quantum information. Classical Kolmogorov complexity is one of the fundamental quantities of classical information theory. In this field, its importance lies in the fact that Kolmogorov complexity is a measure of the information content, and thus compressibility, of a single classical object. The applications of classical Kolmogorov complexity, though, reach beyond classical information theory, to fields such as economics, logic, and other branches of physics. Consequently, not only we can consider the development of a complete theory of quantum Kolmogorov complexity as a fundamental step in quantum information theory, but one hopes that it would reveal itself as powerful as its classical counterpart, allowing for applications in many different areas of knowledge. Indeed, in recent years several different definitions for quantum Kolmogorov complexity have been proposed, thus laying the foundations of such a theory. Nevertheless, these studies are at a very early stage, and the first applications of quantum Kolmogorov complexity are limited to few, if promising, examples. In this research plan, we propose to study quantum Kolmogorov complexity with a broad perspective, investigating both its relations with fundamental notions of quantum theory, such as entanglement, and its applications to quantum information, in particular to quantum communication and quantum computation. These studies will contribute to the development of a complete theory of quantum Kolmogorov complexity as well as give new insight in the whole field of quantum information. Moreover, by investigating the concept of complex unitaries and Hamiltonians, we aim at finding new relations between quantum Kolmogorov complexity and quantum dynamics. This is expected to lead to an insight in the study of quantum chaos, and provide with new connections between this field and quantum information theory.
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