I am taking a course in quantum information theory. In this theory, the state of a system is given by a density matrix, while a measurement is completely positive, trace preserving map of the form: Λ = ∑il i >< i l ⊗ Λi , where Λi is completely positive. Can anyone tell me how this is equivalent to the usual definition of a measurement on a quantum state, where the state is collapsed into some eigenstate of the operator measured. Also it seems crucial to quantum information theory that quantum channels are completely positive and trace preserving. The trace preserving part I get since you want the state of the system to remain and allowed quantum state. But why is the completely positive part needed?