Entropy inequalities and entropic uncertainty relations

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Entropy plays a fundamental role in many areas of physics ranging from condensed matter physics to cosmology. Recent developments in quantum information theory have improved the understanding of entropy by providing operational interpretations of, for instance, von Neumann entropy as data compression rate. Inequalities that relate the entropies of parts of a system are the among the most elementary properties of entropy functions, a well-known example is the strong subadditivity of von Neumann entropy. Inequalities that relate the entropies of measurements of complementary observables, so-called entropic uncertainty relations, first considered in the context of the foundations of physics, have been successfully employed in security proofs of quantum cryptography. This project will study inequalities for the von Neumann entropy and more generally for quantum Rényi entropies. This study has two main goals: First, to decide whether or not there exist further inequalities of the von Neumann entropy (all known inequalities reduce to strong subadditivity). Second, to generalise the entropic uncertainty relations to uncertainty relations for conditional quantum Rényi entropies. The proposed research is of significance to the study of entropy limits and entanglement entropies in spin systems, and will directly impact the study of quantum information theory, cryptography and the foundations of quantum mechanics.