  The concept of entanglement and separability of quantum states is relevant
for several fields in physics. Still, there is a lack of effective operational
methods to characterise these features. We propose a method to certify quantum
separability of two- and multiparticle quantum systems based on an adaptive
polytope approximation. This leads to an algorithm which, for practical
purposes, conclusively recognises two-particle separability for small and
medium-size dimensions. For multiparticle systems, the approach allows to
characterise full separability for up to five qubits or three qutrits; in
addition, different classes of entanglement can be distinguished. Finally, our
methods allow to identify systematically quantum states with interesting
entanglement properties, such as maximally robust states which are separable
for all bipartitions, but not fully separable.
