Final Report Summary - ENFOQI (Entanglement and the foundations of quantum information) Quantum information theory is a relatively new and interdisciplinary field of research, combining quantum mechanics and information theory. This connection has led to deep insights in quantum mechanics and information processing and may further lead to possible practical applications. Examples are quantum cryptography, i.e. the possibility of unconditionally secure communication based on the laws of physics; or quantum computers, which, at least in principle, can solve certain problems more efficiently than classical computers do. Besides these tasks, other interesting applications have been developed, e.g. the possibility of quantum-enhanced high precision metrology or the idea of a quantum simulation of certain physical systems, which are difficult to simulate on a classical computer. It is natural to ask which quantum effects are relevant for the quantum advantage. It is believed that one of these effects is quantum entanglement – the possibility that two quantum systems are so strongly correlated that a description via individual systems does not make sense anymore. Indeed, it has been shown that entanglement is a necessary precondition for certain schemes of quantum cryptography or for phase estimation beyond the classical shot-noise limit, but the role of entanglement is in general not completely clear. Moreover, the characterization of entanglement itself is a difficult and in general unsolved problem, especially in the case of several particles. The advent of quantum information processing has also triggered many developments and experiments on the foundations of quantum mechanics. There, effects like the non-locality of quantum mechanics have been studied. It turned out that the violation of Bell inequalities does not only show the incompatibility of quantum mechanics with local hidden variable theories, but is also connected to quantum advantages in distributed computation scenarios. In the Marie-Curie-Career-Integration-Grant project „Entanglement and the foundations of quantum mechanics“ several aspects of the foundations of quantum mechanics and entanglement theory were explored. There are three fields of research within this project:(a) Entanglement theory: In our work, we mainly investigated entanglement between three or more particles. We proposed and studied a criterion for genuine multiparticle entanglement (the so-called PPT mixture approach) and showed that it is the strongest criterion so far. For many cases, it even solves the problem of deciding whether a quantum state is entangled or not. Moreover, we identified novel classes of multiparticle entangled states (so-called hypergraph states) and characterized their entanglement properties. In addition, we showed that they violate Bell inequalities and are useful resources for quantum metrology. Finally, we considered the notion of steering as a generalization of entanglement, and studied its relation to bound entanglement and the compatibility of measurements.(b) Foundations of quantum mechanics: Here, we mainly worked on the Kochen-Specker theorem, which is a famous result from the sixties of the last century. In short, it states that quantum mechanics cannot be reconciled with classical models that are noncontextual for compatible observables. Here, compatible observables are observables that can be measured simultaneously or in any order without disturbance, and noncontextuality means that the value of an observable does not depend on which other compatible observable is measured jointly with it. In our research, we proposed novel and optimal inequalities to distinguish between quantum mechanics and classical physics. We characterized the maximal violations achievable in quantum mechanics. We also addressed possible loopholes in experiments. Finally, we clarified how the effects of contextuality can be observed in continuous variable systems and how contextuality can be used to test the dimension of quantum systems.(c) Statistical analysis of quantum experiments: Current experiments can observe entanglement between up to 14 particles, but typically only few measurements on the prepared quantum state can be made. Therefore, a careful statistical analysis is required. In our work we showed how state reconstruction schemes like the maximum-likelihood method lead to systematic errors in the reconstructed state. In addition, we worked with experimental groups on the comparison of different state reconstruction schemes and their applicability in practice. The Marie Curie Career Integration Grant helped significantly to build up and maintain the Theoretical Quantum Optics group in Siegen/Germany. It allowed to employ researchers on a short-time basis and it was used to organize workshops in Siegen. Moreover, it helped to integrate our group into the German academic landscape by attracting scientific visitors and allowing us to participate in workshops and conferences. More description can be found on our webpage, http://www.physik.uni-siegen.de/tqo
