Multipartite-entanglement detection with projective measurements

For a projective measurement, the Born rule provides the probability for an outcome in terms of the inner product between a projector and a quantum state. If the projector represents a pure entangled state and the state for a composite system is separable, then we cannot get probability 1 for the outcome. This insight delivers a single condition for entanglement detection. By applying local unitary transformations from the Clifford group, we turn one condition into many. Furthermore, we present two equivalent schemes---one employs global and other requires local projective measurements---to test these conditions in an experiment. Here, a global measurement is characterized by an orthonormal basis that holds local-unitary-equivalent entangled kets. Whereas a local-measurement setting is specified by mutually unbiased bases assigned to the subsystems. We also supply a straightforward (computer) algorithm to generate all the conditions and then to check whether a state is shown entangled or not by these conditions. Finally, we demonstrate every element of our schemes by considering several well-known examples of entangled kets and states.