In my talk, I present results on three relevant problems concerning the characterization of multiparticle entanglement. First, in many experiments one measures certain observables in order to determine the quantum state. The resulting state, however, has often unphysical properties (such as negative eigenvalues), which might be due to systematic errors or due to statistical fluctuations when only a finite number of experiments has been performed. I will introduce a method to distinguish such statistical errors from systematic errors and apply the method to data obtained in a ion-trap experiment [1].

Second, if measurement data without systematic errors are given, the task remains to construct a density matrix. I will show that the freqently used methods of maximum likelihood reconstruction and free least squares optimization lead to a systematic bias, underestimating the fidelity and overestimating the entanglement. This is shown to be a fundamental problem for any state reconstruction scheme that results always in valid density operators [2].

Finally, if in an experiment the quantum state has been reconstructed properly, the question remains to show that the state is entangled. For that, I will introduce a method to characterize genuine multipartite entanglement using a multipartite extension of the criterion of the positivity of the partial transpose [3]. This criterion can simply be evaluated using semidefinite programming, even if the density matrix is not completely known.

[1] T. Moroder et al., Phys. Rev. Lett. 110, 180401 (2013)

[2] C. Schwemmer et al., arXiv:tomo.rrow

[3] B. Jungnitsch et al., Phys. Rev. Lett. 106, 190502 (2011)