  Characterizing entanglement is central for quantum information science.
Special observables which indicate entanglement, so-called entanglement
witnesses, are a widely used tool for this task. The construction of these
witnesses typically relies on the observation that quantum states with a high
fidelity to some entangled target state are entangled, too. We introduce a
general method to construct entanglement witnesses based on the Schmidt
decomposition of observables. The method works for two- and, more importantly,
many-body systems and is strictly stronger than fidelity-based constructions.
The resulting witnesses can also be used to quantify entanglement as well as to
characterize the dimensionality of it. Finally, we present experimentally
relevant examples, where our approach improves entanglement detection
significantly.
