Common tools for obtaining physical density matrices in experimental quantum state tomography are shown here to cause systematic errors. For example, using maximum likelihood or least squares optimization for state reconstruction, we observe a systematic underestimation of the fidelity and an overestimation of entanglement. A solution for this problem can be achieved by a linear evaluation of the data yielding reliable and computational simple bounds including error bars.

It is argued that a weak value of an observable is a robust property of a single pre-and postselected quantum system rather than a statistical property. During an infinitesimal time a system with a given weak value affects other systems as if it were in an eigenstate with eigenvalue equal to the weak value. This differs significantly from the action of a system pre-selected only and possessing a numerically equal expectation value. The weak value has a physical meaning beyond a conditional average of a pointer in the weak measurement procedure. The difference between the weak value and the expectation value has been demonstrated on the example of photon polarization. In addition, the weak values for systems pre-and post-selected in mixed states are considered.

Non-classical correlations between measurement results make entanglement the essence of quantum physics and the main resource for quantum information applications. Surprisingly, there are n-particle states which do not exhibit n-partite correlations at all but still are genuinely n-partite entangled. We introduce a general construction principle for such states, implement them in a multiphoton experiment and analyze their properties in detail. Remarkably, even without n-partite correlations, these states do violate Bell inequalities showing that there is no classical, i.e., local realistic model describing their properties.PACS numbers: 03.67. Mn, 03.65.Ud Correlations between measurement results are the most prominent feature of entanglement. They made Einstein, Podolski and Rosen [1] to question the completeness of quantum mechanics, and are nowadays the main ingredient for the many applications of quantum information like entanglement based quantum key distribution [2] or quantum teleportation [3].Correlations enable us, e.g., when observing two maximally entangled qubits, to use a measurement result observed on the first system to infer exactly the measurement result on the second system. In this scenario the two particle correlations are formally given by the expectation value of the product of the measurement results obtained by the two observers. Note, the single particle correlation, i.e., the expectation value of the results for one or the other particle are zero in this case. Consequently, we cannot predict anything about the individual results. When studying the entanglement between n particles, a natural extension is to consider n-partite correlations, i.e., the expectation value of the product of n measurement results. Such correlation functions are frequently used in classical statistics and signal analysis [4], moreover in quantum information almost all standard tools for analyzing n-partite systems like multi-party entanglement witnesses [5,6] and Bell inequalities [7,8] are based on the n-partite correlation functions.Recently, Kaszlikowski et al. [9] pointed at a particular quantum state with vanishing multi-party correlations which, however, is genuinely multipartite entangled. This discovery, of course, prompted vivid discussions on a viable definition of classical and quantum correlations [10,11]. Still, the question remains what makes up such * tomasz.paterek@ntu.edu.sg states with no full n-partite correlations and how nonclassical they can be, i.e., whether they are not only entangled but whether they also violate a Bell inequality.Here, we generalize, highlight and experimentally test such remarkable quantum states. We introduce a simple principle how to construct states without n-partite correlations for odd n and show that there are infinitely many such states which are genuinely n-partite entangled. We implement three and five qubit no-correlation states in a multiphoton experiment and demonstrate that these states do not exhibit n-partite correlations. Yet, due to the existence o...