Characterizing quantum correlations : entanglement, uncertainty relations and exponential families

This thesis is concerned with different characterizations of multi-particle quantum correlations and with entropic uncertainty relations.

The effect of statistical errors on the detection of entanglement is investigated. First, general results on the statistical significance of entanglement witnesses are obtained. Then, using an error model for experiments with polarization-entangled photons, it is demonstrated that Bell inequalities with lower violation can have higher significance.

The question for the best observables to discriminate between a state and the equivalence class of another state is addressed. Two measures for the discrimination strength of an observable are defined, and optimal families of observables are constructed for several examples.

A property of stabilizer bases is shown which is a natural generalization of mutual unbiasedness. For sets of several dichotomic, pairwise anticommuting observables, uncertainty relations using different entropies are constructed in a systematic way.

Exponential families provide a classification of states according to their correlations. In this classification scheme, a state is considered as k-correlated if it can be written as thermal state of a k-body Hamiltonian. Witness operators for the detection of higher-order interactions are constructed, and an algorithm for the computation of the nearest k-correlated state is developed.