For interacting (or open) systems, such as those undergoing measurement, the situation is entirely different. To begin with, the state changes experienced by such systems cannot be accounted for exclusively by a transformation on the set of pure states (that is, those associated to vectors of norm 1 in H). After such an interaction, a system in a pure state φ may no longer be in the pure state φ. In general it will be in a statistical mix of a sequence of pure states φ 1,..., φ k with respective probabilities λ 1,..., λ k. The transition from a pure state to a mixed state is known as decoherence.