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.