where the are positively valued probabilities and the formula_29's and formula_30's are themselves mixed states (density operators) on the subsystems and respectively. In other words, a state is separable if it is a probability distribution over uncorrelated states, or product states. By writing the density matrices as sums of pure ensembles and expanding, we may assume without loss of generality that formula_29 and formula_30 are themselves pure ensembles. A state is then said to be entangled if it is not separable.