Multipartite entanglement

Since the works of Einstein and Schrödinger, entanglement is known as a mysterious phenomenon in quantum mechanics. Roughly speaking, a tow or more particles are entangled, if the properties of the total system are not included in the properties of the single systems. This leads to strong correlations between the particles, for which Einstein coined the term "spooky action at a distance". But Einstein as well as Schrödinger considered entanglement as a strange phenomenon, which shows that quantum mechanics is not a proper theory. However, the view on entanglement has changed dramatically in the last years. It has been shown that entanglement can be viewed as as resource, which enables tasks which are impossible without it. For instance, entangled particles can be used to perform quantum cryptography. Furthermore one can, provided many particles have been entangled in a certain state, perform a computation with measurements only. This is the so called one- way quantum computer. Due to all these discoveries, entanglement is now under intensive research, theoretically as well as experimentally. From the theoretical side, many task have been recognized, where entanglement helps to find a solution. From the experimental side, many attempts have been undertaken, in order to entangle two or more particles. Indeed, entanglement has been observed for up to eight ions or six photons. Despite of these attempts, entanglement is by no means understood. This is especially the case for more than two particles. Here, different classes of entanglement exists, but it is not easy to decide to which class a given state belongs. This project is aimed at a detailed investigation of multipartite entanglement. First, we will develop efficient criteria for the discrimination of the different classes of entanglement. Further, we will study so-called entanglement measures, which aim at a quantification of entanglement. But also experimental aspects of multipartite entanglement will be studied. First, we will develop efficient tools for the analysis of multipartite entanglement in experiments. This will enable new experiments, since the strict verification of multipartite entanglement is a crucial problem in nowadays experiments. Finally, we will investigate certain models of condensed matter physics. At low temperatures entanglement arises naturally in these systems. Here we will clarify to which extent one can gain new insight into condensed matter physics from the analysis of the multipartite entanglement properties.

Since the thirties of last century entanglement is known as a mysterious phenomenon in quantum mechanics. Roughly speaking, two particles are called entangled if the properties of the entire system cannot be derived from those of the individual systems. Due to the advancement of quantum technology, entanglement can now be observed experimentally with ions or photons. The theoretical understanding, however, is still not complete, especially for the case of more than two particles. In fact, many problems are completely unsolved. In this START project, significant progress has been made concerning the theoretical understanding of entanglement as well as the experimental analysis of it. First, we discussed the question whether for a quantum state of several particles all particles are entangled with each other, or whether only a subset is entangled. We presented mathematical criteria for entanglement that are more efficient than all previously known criteria. It has even been shown that these criteria solve the problem for most interesting cases. Another key question is how to detect entanglement in experiments. In experiments one often cannot determine the complete quantum state, and one can only measure few observables. We proposed a method to detect the entanglement with high statistical significance and this method was implemented in collaboration with experimentalists. In another study it was shown how to distinguish statistical fluctuations from systematic errors in experiments. Furthermore, during the project we had many collaborations with experimental groups concerning the analysis of their measured data. For example, our theoretical results were used in the first experiment where ten-qubit entanglement was observed. In addition, we developed the analysis tools for the first experiment which observed bound entanglement with ions. Finally, we obtained important results in connection to the Kochen-Specker theorem. This theorem states that quantum mechanics is incompatible with the assumption of non-contextuality. We provided the theory behind the first experiment in which this contradiction has been observed in a state independent manner experimentally. In addition, we derived new inequalities that can be used to verify the Kochen-Specker theorem experimentally. In summary, the results of the START project led to a significantly improved understanding of multiparticle entanglement and allowed novel and ground-breaking experiments. In addition, fundamental results have been obtained in the research field of quantum contextuality and it can be expected that this field will lead to many new results in the future.