  We discuss the problem of determining whether the state of several quantum
mechanical subsystems is entangled. As in previous work on two subsystems we
introduce a procedure for checking separability that is based on finding state
extensions with appropriate properties and may be implemented as a semidefinite
program. The main result of this work is to show that there is a series of
tests of this kind such that if a multiparty state is entangled this will
eventually be detected by one of the tests. The procedure also provides a means
of constructing entanglement witnesses that could in principle be measured in
order to demonstrate that the state is entangled.
