  We derive a family of necessary separability criteria for finite-dimensional
systems based on inequalities for variances of observables. We show that every
pure bipartite entangled state violates some of these inequalities.
Furthermore, a family of bound entangled states and true multipartite entangled
states can be detected. The inequalities also allow to distinguish between
different classes of true tripartite entanglement for qubits. We formulate an
equivalent criterion in terms of covariance matrices. This allows us to apply
criteria known from the regime of continuous variables to finite-dimensional
systems.
