  We show how to design families of operational criteria that distinguish
entangled from separable quantum states. The simplest of these tests
corresponds to the well-known Peres-Horodecki positive partial transpose (PPT)
criterion, and the more complicated tests are strictly stronger. The new
criteria are tractable due to powerful computational and theoretical methods
for the class of convex optimization problems known as semidefinite programs.
We successfully applied the results to many low-dimensional states from the
literature where the PPT test fails. As a byproduct of the criteria, we provide
an explicit construction of the corresponding entanglement witnesses.
