Final Report Summary - GEDENTQOPT (Generation and detection of many-particle entanglement in quantum optical systems) Quantum Information Science has become one of the major research fields in physics, to a large extent due to the recent breakthroughs in manipulating the quantum states of cold atoms, trapped ions, and photons. As experimentalists create larger and larger coherent quantum systems, new theoretical methods are needed. On the one hand, meaningful goals must be set for the quantum experiments, by determining the quantum states that are useful for quantum information processing applications. On the other hand, new methods are needed for the verification of the experimentally created state, since full tomography of the quantum state is not possible for state-of-the-art system sizes. During the last decade, quantum entanglement has been intensively studied within quantum information science and has also appeared as a natural goal of recent quantum experiments. Because of that the theoretical background of detecting entanglement has been rapidly developing. However, most of this development concentrated on bipartite or few-party entanglement, while today's experiments typically involve many particles. Thus, as one of the most interesting part of quantum optics and quantum information, I chose to study multi-partite entanglement theory, with a stress on creation and generation of many-particle entanglement. As planned in the grant proposal, with external collaborators, we developed Permutationally Invariant Quantum State Tomography, a method for scalable tomography of multipartite quantum systems. The number of measurements needed scale polynomially with the system size, while the scaling is exponential for full state tomography. Our method has been tested in a four-qubit experiment and in a six-qubit experiment with photons, creating Dicke states, in the group of H. Weinfurter in the Max Planck Institute for Quantum Optics in Munich. Our tomographic procedure is one of the few possible choices in many-particle systems with an individual access to the particles, for example, in trapped cold ion quantum registers, photonic systems or, in the near future, optical lattice quantum registers. The above collaboration yielded further results on quantum tomography, concerning the long standing problem of handling the negative eigenvalues of the density matrix obtained experimentally. In the second part of the project, we considered many-particle systems in which only collective observables can be measured. We determined the full set of spin squeezing inequalities for ensembles of particles with a spin larger than 1/2. Our entanglement conditions are very useful, since most of the experiments are done with such particles. Our set of conditions is a complete set in the sense than no new conditions can be found that detect more entangled states than ours in the large particle number limit. As recent experiments focus more and more on entanglement in cold atomic ensembles, we hope that these inequalities will be used to detect entanglement in the vicinity of important quantum states such as singlets, Dicke states and planar squeezed states. Connected to one of the entanglement conditions mentioned above, in collaboration with M. W. Mitchell (ICFO, Barcelona), we proposed a spin squeezing procedure, based on a spin squeezing of all the three collective spin observables, that creates a many-body singlet state. We also considered the use of this state for gradient magnetometry and the related problem of modeling the dynamics of large atomic systems analytically. The squeezing of all the three spin components, together with a necessary feedback step followed by incoherent pumping, has been realized experimentally in cold atoms in the group of M. W. Mitchell.We also developed entanglement conditions that can detect the depth of entanglement close to Dicke states with spin-squeezing inequalities. These have been applied in the cold gas experiments of the group of Prof. C. Klempt at the University of Hannover, Germany. 28-particle entanglement has been detected in an 8000 atoms. Dicke states are quantum states very promising metrologically, since they are more robust to particle loss than Greenberger-Horne-Zeilinger (GHZ) states, while they still make possible to reach, apart from a constant factor, the Heisenberg limit.The third part of the project was about theoretical issues in quantum metrology. We succeeded to connect multipartite entanglement to quantum metrology and show that genuine multipartite entanglement is needed to achieve the maximum precision in very general metrological tasks in a linear interferometer. This was a missing piece in understanding why this type of entanglement is important. Connected to these findings, experiments have been carried out that detected the entanglement of a quantum state based on the metrological performance of the state in cold atoms (group of C. Klempt, Hannover), and detected multipartite entanglement in photons also based on the metrological usefulness of the quantum state (group of H. Weinfurter, Munich).We also showed that the quantum Fisher information, a fundamental quantity in quantum metrology, is the convex roof of the variance. This is a very important result, since convex roofs so far appeared only concerning entanglement measures. We used this fact to develop methods to estimate the quantum Fisher information based on few measurements. We also worked out an efficient method to obtain convex roofs numerically.
