  Verification of entanglement is an important tool to characterize sources and
devices for use in quantum computing and communication applications. In a vast
majority of experiments entanglement witnesses (EW) are used in order to prove
the presence of entanglement in a quantum state. EWs can be constructed from
available measurement results and do not require a reconstruction of the whole
density matrix (full tomography), which is especially valuable for
high-dimensional systems. We provide a method to construct {\it accessible
nonlinear EWs}, which incorporate two important properties. First, they improve
on linear EWs in the sense that each non-linear EW detects more entangled
states than its linear counterpart and therefore allow the verification of
entanglement without critical dependence on having found the 'right' linear
witness. Second, they can be evaluated using exactly the same data as for the
evaluation of the original linear witness. This allows a reanalysis of
published experimental data to strengthen statements about entanglement
verification without the requirement to perform additional measurements. These
particular properties make the accessible nonlinear EWs attractive for the
implementations in current experiments, for they can also enhance the
statistical significance of the entanglement verification.
